Database for the paper "Binary optimal linear codes with various hull dimensions"
Part 1
Exhaustive search for k <= n <= 12 (when n=12, k <=4)
Notation:
di means highest minimum distance of an [n,k,d] code with hull dimension h=i (0 <= i <=5)
##################################
n,k 2 1
d0: 1
[2, 1, 1] Linear Code over GF(2)
Generator matrix:
[1 0]
d1: 2
[2, 1, 2] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1]
d2: 0
[2, 0, 2] Cyclic Linear Code over GF(2)
d3: 0
[2, 0, 2] Cyclic Linear Code over GF(2)
d4: 0
[2, 0, 2] Cyclic Linear Code over GF(2)
d5: 0
[2, 0, 2] Cyclic Linear Code over GF(2)
d6: 0
[2, 0, 2] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 2 1
h0, h1, h2, h3, h4, h5, h6: 1 2 0 0 0 0 0
##################################
n,k 2 2
d0: 1
[2, 2, 1] Cyclic Linear Code over GF(2)
d1: 0
[2, 0, 2] Cyclic Linear Code over GF(2)
d2: 0
[2, 0, 2] Cyclic Linear Code over GF(2)
d3: 0
[2, 0, 2] Cyclic Linear Code over GF(2)
d4: 0
[2, 0, 2] Cyclic Linear Code over GF(2)
d5: 0
[2, 0, 2] Cyclic Linear Code over GF(2)
d6: 0
[2, 0, 2] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 2 2
h0, h1, h2, h3, h4, h5, h6: 1 0 0 0 0 0 0
##################################
n,k 3 1
d0: 3
[3, 1, 3] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1 1]
d1: 2
[3, 1, 2] Linear Code over GF(2)
Generator matrix:
[1 1 0]
d2: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d3: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d4: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d5: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d6: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 3 1
h0, h1, h2, h3, h4, h5, h6: 3 2 0 0 0 0 0
##################################
n,k 3 2
d0: 2
[3, 2, 2] Cyclic Linear Code over GF(2)
Generator matrix:
[1 0 1]
[0 1 1]
d1: 1
[3, 2, 1] Linear Code over GF(2)
Generator matrix:
[1 0 1]
[0 1 0]
d2: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d3: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d4: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d5: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d6: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 3 2
h0, h1, h2, h3, h4, h5, h6: 2 1 0 0 0 0 0
##################################
n,k 3 3
d0: 1
[3, 3, 1] Cyclic Linear Code over GF(2)
d1: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d2: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d3: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d4: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d5: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
d6: 0
[3, 0, 3] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 3 3
h0, h1, h2, h3, h4, h5, h6: 1 0 0 0 0 0 0
##################################
n,k 4 1
d0: 3
[4, 1, 3] Linear Code over GF(2)
Generator matrix:
[1 1 1 0]
d1: 4
[4, 1, 4] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1 1 1]
d2: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d3: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d4: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d5: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d6: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 4 1
h0, h1, h2, h3, h4, h5, h6: 3 4 0 0 0 0 0
##################################
n,k 4 2
d0: 2
[4, 2, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0]
[0 1 1 0]
d1: 1
[4, 2, 1] Linear Code over GF(2)
Generator matrix:
[1 0 1 0]
[0 1 0 0]
d2: 2
[4, 2, 2] Quasicyclic of degree 2 Linear Code over GF(2)
Generator matrix:
[1 0 0 1]
[0 1 1 0]
d3: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d4: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d5: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d6: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 4 2
h0, h1, h2, h3, h4, h5, h6: 2 1 2 0 0 0 0
##################################
n,k 4 3
d0: 1
[4, 3, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
d1: 2
[4, 3, 2] Cyclic Linear Code over GF(2)
Generator matrix:
[1 0 0 1]
[0 1 0 1]
[0 0 1 1]
d2: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d3: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d4: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d5: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d6: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 4 3
h0, h1, h2, h3, h4, h5, h6: 1 2 0 0 0 0 0
##################################
n,k 4 4
d0: 1
[4, 4, 1] Cyclic Linear Code over GF(2)
d1: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d2: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d3: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d4: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d5: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
d6: 0
[4, 0, 4] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 4 4
h0, h1, h2, h3, h4, h5, h6: 1 0 0 0 0 0 0
##################################
n,k 5 1
d0: 5
[5, 1, 5] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1]
d1: 4
[5, 1, 4] Linear Code over GF(2)
Generator matrix:
[1 1 1 1 0]
d2: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d3: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d4: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d5: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d6: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 5 1
h0, h1, h2, h3, h4, h5, h6: 5 4 0 0 0 0 0
##################################
n,k 5 2
d0: 2
[5, 2, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0]
[0 1 1 0 0]
d1: 3
[5, 2, 3] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 1]
[0 1 1 1 0]
d2: 2
[5, 2, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0]
[0 1 1 0 0]
d3: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d4: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d5: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d6: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 5 2
h0, h1, h2, h3, h4, h5, h6: 2 3 2 0 0 0 0
##################################
n,k 5 3
d0: 2
[5, 3, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 1]
[0 1 0 1 0]
[0 0 1 1 0]
d1: 2
[5, 3, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0]
[0 1 0 1 0]
[0 0 1 1 0]
d2: 1
[5, 3, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1]
[0 1 0 1 0]
[0 0 1 0 0]
d3: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d4: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d5: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d6: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 5 3
h0, h1, h2, h3, h4, h5, h6: 2 2 1 0 0 0 0
##################################
n,k 5 4
d0: 2
[5, 4, 2] Cyclic Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1]
[0 1 0 0 1]
[0 0 1 0 1]
[0 0 0 1 1]
d1: 1
[5, 4, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
d2: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d3: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d4: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d5: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d6: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 5 4
h0, h1, h2, h3, h4, h5, h6: 2 1 0 0 0 0 0
##################################
n,k 5 5
d0: 1
[5, 5, 1] Cyclic Linear Code over GF(2)
d1: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d2: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d3: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d4: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d5: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
d6: 0
[5, 0, 5] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 5 5
h0, h1, h2, h3, h4, h5, h6: 1 0 0 0 0 0 0
##################################
n,k 6 1
d0: 5
[6, 1, 5] Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 0]
d1: 6
[6, 1, 6] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1]
d2: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d3: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d4: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d5: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d6: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 6 1
h0, h1, h2, h3, h4, h5, h6: 5 6 0 0 0 0 0
##################################
n,k 6 2
d0: 3
[6, 2, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1]
[0 1 1 1 0 0]
d1: 3
[6, 2, 3] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 1 0]
[0 1 1 1 0 0]
d2: 4
[6, 2, 4] Quasicyclic of degree 3 Linear Code over GF(2)
Generator matrix:
[1 0 1 1 0 1]
[0 1 1 1 1 0]
d3: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d4: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d5: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d6: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 6 2
h0, h1, h2, h3, h4, h5, h6: 3 3 4 0 0 0 0
##################################
n,k 6 3
d0: 2
[6, 3, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 1 0]
[0 1 0 1 0 0]
[0 0 1 1 0 0]
d1: 2
[6, 3, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 0]
[0 1 0 1 0 0]
[0 0 1 1 0 0]
d2: 3
[6, 3, 3] Quasicyclic of degree 2 Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1]
[0 1 0 1 0 1]
[0 0 1 1 1 0]
d3: 2
[6, 3, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1]
[0 1 0 0 1 0]
[0 0 1 1 0 0]
d4: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d5: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d6: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 6 3
h0, h1, h2, h3, h4, h5, h6: 2 2 3 2 0 0 0
##################################
n,k 6 4
d0: 2
[6, 4, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0]
[0 1 0 0 1 0]
[0 0 1 0 1 0]
[0 0 0 1 1 0]
d1: 1
[6, 4, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
d2: 2
[6, 4, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1]
[0 1 0 0 1 0]
[0 0 1 0 1 0]
[0 0 0 1 1 0]
d3: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d4: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d5: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d6: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 6 4
h0, h1, h2, h3, h4, h5, h6: 2 1 2 0 0 0 0
##################################
n,k 6 5
d0: 1
[6, 5, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
d1: 2
[6, 5, 2] Cyclic Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1]
[0 1 0 0 0 1]
[0 0 1 0 0 1]
[0 0 0 1 0 1]
[0 0 0 0 1 1]
d2: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d3: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d4: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d5: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d6: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 6 5
h0, h1, h2, h3, h4, h5, h6: 1 2 0 0 0 0 0
##################################
n,k 6 6
d0: 1
[6, 6, 1] Cyclic Linear Code over GF(2)
d1: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d2: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d3: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d4: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d5: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
d6: 0
[6, 0, 6] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 6 6
h0, h1, h2, h3, h4, h5, h6: 1 0 0 0 0 0 0
##################################
n,k 7 1
d0: 7
[7, 1, 7] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1]
d1: 6
[7, 1, 6] Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 0]
d2: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d3: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d4: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d5: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d6: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 7 1
h0, h1, h2, h3, h4, h5, h6: 7 6 0 0 0 0 0
##################################
n,k 7 2
d0: 4
[7, 2, 4] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 1 1]
[0 1 1 1 1 0 0]
d1: 4
[7, 2, 4] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 0 1 1]
[0 1 1 1 1 0 0]
d2: 4
[7, 2, 4] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 0 1 0]
[0 1 1 1 1 0 0]
d3: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d4: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d5: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d6: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 7 2
h0, h1, h2, h3, h4, h5, h6: 4 4 4 0 0 0 0
##################################
n,k 7 3
d0: 3
[7, 3, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 1]
[0 1 0 1 0 1 0]
[0 0 1 1 1 0 0]
d1: 3
[7, 3, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 1]
[0 1 0 1 0 1 0]
[0 0 1 1 1 0 0]
d2: 3
[7, 3, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 0]
[0 1 0 1 0 1 0]
[0 0 1 1 1 0 0]
d3: 4
[7, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 1]
[0 1 0 1 1 0 1]
[0 0 1 1 1 1 0]
d4: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d5: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d6: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 7 3
h0, h1, h2, h3, h4, h5, h6: 3 3 3 4 0 0 0
##################################
n,k 7 4
d0: 2
[7, 4, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0]
[0 1 0 0 1 0 0]
[0 0 1 0 1 0 0]
[0 0 0 1 1 0 0]
d1: 2
[7, 4, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1]
[0 1 0 0 1 0 0]
[0 0 1 0 1 0 0]
[0 0 0 1 1 0 0]
d2: 2
[7, 4, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0]
[0 1 0 0 1 0 0]
[0 0 1 0 1 0 0]
[0 0 0 1 1 0 0]
d3: 3
[7, 4, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 1]
[0 1 0 0 0 1 1]
[0 0 1 0 1 0 1]
[0 0 0 1 1 1 0]
d4: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d5: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d6: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 7 4
h0, h1, h2, h3, h4, h5, h6: 2 2 2 3 0 0 0
##################################
n,k 7 5
d0: 2
[7, 5, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1]
[0 1 0 0 0 1 0]
[0 0 1 0 0 1 0]
[0 0 0 1 0 1 0]
[0 0 0 0 1 1 0]
d1: 2
[7, 5, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0]
[0 1 0 0 0 1 0]
[0 0 1 0 0 1 0]
[0 0 0 1 0 1 0]
[0 0 0 0 1 1 0]
d2: 1
[7, 5, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1]
[0 1 0 0 0 1 0]
[0 0 1 0 0 0 0]
[0 0 0 1 0 0 0]
[0 0 0 0 1 0 0]
d3: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d4: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d5: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d6: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 7 5
h0, h1, h2, h3, h4, h5, h6: 2 2 1 0 0 0 0
##################################
n,k 7 6
d0: 2
[7, 6, 2] Cyclic Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1]
[0 1 0 0 0 0 1]
[0 0 1 0 0 0 1]
[0 0 0 1 0 0 1]
[0 0 0 0 1 0 1]
[0 0 0 0 0 1 1]
d1: 1
[7, 6, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1]
[0 1 0 0 0 0 0]
[0 0 1 0 0 0 0]
[0 0 0 1 0 0 0]
[0 0 0 0 1 0 0]
[0 0 0 0 0 1 0]
d2: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d3: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d4: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d5: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d6: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 7 6
h0, h1, h2, h3, h4, h5, h6: 2 1 0 0 0 0 0
##################################
n,k 7 7
d0: 1
[7, 7, 1] Cyclic Linear Code over GF(2)
d1: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d2: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d3: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d4: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d5: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
d6: 0
[7, 0, 7] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 7 7
h0, h1, h2, h3, h4, h5, h6: 1 0 0 0 0 0 0
##################################
n,k 8 1
d0: 7
[8, 1, 7] Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1 0]
d1: 8
[8, 1, 8] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1 1]
d2: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d3: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d4: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d5: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d6: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 8 1
h0, h1, h2, h3, h4, h5, h6: 7 8 0 0 0 0 0
##################################
n,k 8 2
d0: 5
[8, 2, 5] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 0 0 1 1]
[0 1 1 1 1 1 0 0]
d1: 4
[8, 2, 4] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 0 1 1 0]
[0 1 1 1 1 0 0 0]
d2: 4
[8, 2, 4] Quasicyclic of degree 4 Linear Code over GF(2)
Generator matrix:
[1 0 1 1 0 1 0 0]
[0 1 1 1 1 0 0 0]
d3: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d4: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d5: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d6: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 8 2
h0, h1, h2, h3, h4, h5, h6: 5 4 4 0 0 0 0
##################################
n,k 8 3
d0: 3
[8, 3, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 1 0]
[0 1 0 1 0 1 0 0]
[0 0 1 1 1 0 0 0]
d1: 4
[8, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 0 1]
[0 1 0 1 1 0 1 0]
[0 0 1 1 1 1 0 0]
d2: 4
[8, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 1 1]
[0 1 0 1 1 0 1 0]
[0 0 1 1 1 1 0 0]
d3: 4
[8, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 1 0]
[0 1 0 1 1 0 1 0]
[0 0 1 1 1 1 0 0]
d4: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d5: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d6: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 8 3
h0, h1, h2, h3, h4, h5, h6: 3 4 4 4 0 0 0
##################################
n,k 8 4
d0: 3
[8, 4, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 1]
[0 1 0 0 0 1 0 1]
[0 0 1 0 1 0 1 0]
[0 0 0 1 1 1 0 0]
d1: 3
[8, 4, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0 1]
[0 1 0 0 0 1 1 0]
[0 0 1 0 1 0 1 0]
[0 0 0 1 1 1 0 0]
d2: 3
[8, 4, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 1 1]
[0 1 0 0 0 1 1 0]
[0 0 1 0 1 0 1 0]
[0 0 0 1 1 1 0 0]
d3: 3
[8, 4, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 1 0]
[0 1 0 0 0 1 1 0]
[0 0 1 0 1 0 1 0]
[0 0 0 1 1 1 0 0]
d4: 4
[8, 4, 4] Quasicyclic of degree 2 Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 0 1 1 1 1 0]
d5: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d6: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 8 4
h0, h1, h2, h3, h4, h5, h6: 3 3 3 3 4 0 0
##################################
n,k 8 5
d0: 2
[8, 5, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 0]
[0 1 0 0 0 1 0 0]
[0 0 1 0 0 1 0 0]
[0 0 0 1 0 1 0 0]
[0 0 0 0 1 1 0 0]
d1: 2
[8, 5, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 0]
[0 1 0 0 0 1 0 0]
[0 0 1 0 0 1 0 0]
[0 0 0 1 0 1 0 0]
[0 0 0 0 1 1 0 0]
d2: 2
[8, 5, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 1]
[0 1 0 0 0 1 0 1]
[0 0 1 0 0 1 1 0]
[0 0 0 1 0 1 0 0]
[0 0 0 0 1 1 0 0]
d3: 2
[8, 5, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1]
[0 1 0 0 0 0 1 0]
[0 0 1 0 0 1 0 0]
[0 0 0 1 0 1 0 0]
[0 0 0 0 1 1 0 0]
d4: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d5: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d6: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 8 5
h0, h1, h2, h3, h4, h5, h6: 2 2 2 2 0 0 0
##################################
n,k 8 6
d0: 2
[8, 6, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0]
[0 1 0 0 0 0 1 0]
[0 0 1 0 0 0 1 0]
[0 0 0 1 0 0 1 0]
[0 0 0 0 1 0 1 0]
[0 0 0 0 0 1 1 0]
d1: 1
[8, 6, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0]
[0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0]
d2: 2
[8, 6, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1]
[0 1 0 0 0 0 1 0]
[0 0 1 0 0 0 1 0]
[0 0 0 1 0 0 1 0]
[0 0 0 0 1 0 1 0]
[0 0 0 0 0 1 1 0]
d3: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d4: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d5: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d6: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 8 6
h0, h1, h2, h3, h4, h5, h6: 2 1 2 0 0 0 0
##################################
n,k 8 7
d0: 1
[8, 7, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0]
d1: 2
[8, 7, 2] Cyclic Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 1]
[0 0 1 0 0 0 0 1]
[0 0 0 1 0 0 0 1]
[0 0 0 0 1 0 0 1]
[0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 1 1]
d2: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d3: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d4: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d5: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d6: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 8 7
h0, h1, h2, h3, h4, h5, h6: 1 2 0 0 0 0 0
##################################
n,k 8 8
d0: 1
[8, 8, 1] Cyclic Linear Code over GF(2)
d1: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d2: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d3: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d4: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d5: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
d6: 0
[8, 0, 8] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 8 8
h0, h1, h2, h3, h4, h5, h6: 1 0 0 0 0 0 0
##################################
n,k 9 1
d0: 9
[9, 1, 9] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1 1 1]
d1: 8
[9, 1, 8] Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1 1 0]
d2: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d3: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d4: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d5: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d6: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 9 1
h0, h1, h2, h3, h4, h5, h6: 9 8 0 0 0 0 0
##################################
n,k 9 2
d0: 6
[9, 2, 6] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 1 0 0 1 1]
[0 1 1 1 1 1 1 0 0]
d1: 5
[9, 2, 5] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 1 1 1]
[0 1 1 1 1 1 0 0 0]
d2: 4
[9, 2, 4] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 0 1 0 0 0]
[0 1 1 1 1 0 0 0 0]
d3: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d4: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d5: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d6: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 9 2
h0, h1, h2, h3, h4, h5, h6: 6 5 4 0 0 0 0
##################################
n,k 9 3
d0: 4
[9, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 1 0 1 0 1]
[0 1 0 1 0 0 1 1 0]
[0 0 1 1 1 1 0 0 0]
d1: 4
[9, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 0 1 0]
[0 1 0 1 1 0 1 0 0]
[0 0 1 1 1 1 0 0 0]
d2: 4
[9, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 1 1 0]
[0 1 0 1 1 0 1 0 0]
[0 0 1 1 1 1 0 0 0]
d3: 4
[9, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 1 0 0]
[0 1 0 1 1 0 1 0 0]
[0 0 1 1 1 1 0 0 0]
d4: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d5: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d6: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 9 3
h0, h1, h2, h3, h4, h5, h6: 4 4 4 4 0 0 0
##################################
n,k 9 4
d0: 4
[9, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0 1 1]
[0 1 0 0 1 0 1 0 1]
[0 0 1 0 1 1 0 1 0]
[0 0 0 1 1 1 1 0 0]
d1: 3
[9, 4, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0 1 0]
[0 1 0 0 0 1 1 0 0]
[0 0 1 0 1 0 1 0 0]
[0 0 0 1 1 1 0 0 0]
d2: 4
[9, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 0 0 1]
[0 1 0 0 1 0 1 1 0]
[0 0 1 0 1 1 0 1 0]
[0 0 0 1 1 1 1 0 0]
d3: 4
[9, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 1]
[0 1 0 0 1 0 1 1 0]
[0 0 1 0 1 1 0 1 0]
[0 0 0 1 1 1 1 0 0]
d4: 4
[9, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 0]
[0 1 0 0 1 0 1 1 0]
[0 0 1 0 1 1 0 1 0]
[0 0 0 1 1 1 1 0 0]
d5: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d6: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 9 4
h0, h1, h2, h3, h4, h5, h6: 4 3 4 4 4 0 0
##################################
n,k 9 5
d0: 3
[9, 5, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0 1]
[0 1 0 0 0 1 0 0 1]
[0 0 1 0 0 0 1 1 0]
[0 0 0 1 0 1 0 1 0]
[0 0 0 0 1 1 1 0 0]
d1: 3
[9, 5, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 1 1]
[0 1 0 0 0 1 0 0 1]
[0 0 1 0 0 0 1 1 0]
[0 0 0 1 0 1 0 1 0]
[0 0 0 0 1 1 1 0 0]
d2: 3
[9, 5, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 0 1]
[0 1 0 0 0 1 1 1 0]
[0 0 1 0 0 0 1 1 0]
[0 0 0 1 0 1 0 1 0]
[0 0 0 0 1 1 1 0 0]
d3: 3
[9, 5, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 1]
[0 1 0 0 0 0 1 1 1]
[0 0 1 0 0 1 0 0 1]
[0 0 0 1 0 1 0 1 0]
[0 0 0 0 1 1 1 0 0]
d4: 2
[9, 5, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 1 1]
[0 1 0 0 0 0 0 1 1]
[0 0 1 0 0 0 1 0 1]
[0 0 0 1 0 0 1 1 0]
[0 0 0 0 1 1 0 0 0]
d5: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d6: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 9 5
h0, h1, h2, h3, h4, h5, h6: 3 3 3 3 2 0 0
##################################
n,k 9 6
d0: 2
[9, 6, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0 0]
[0 1 0 0 0 0 1 0 0]
[0 0 1 0 0 0 1 0 0]
[0 0 0 1 0 0 1 0 0]
[0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 1 1 0 0]
d1: 2
[9, 6, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 1 0 0]
[0 0 1 0 0 0 1 0 0]
[0 0 0 1 0 0 1 0 0]
[0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 1 1 0 0]
d2: 2
[9, 6, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 0]
[0 1 0 0 0 0 1 0 0]
[0 0 1 0 0 0 1 0 0]
[0 0 0 1 0 0 1 0 0]
[0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 1 1 0 0]
d3: 2
[9, 6, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 1 1]
[0 1 0 0 0 0 0 1 1]
[0 0 1 0 0 0 1 0 1]
[0 0 0 1 0 0 1 1 0]
[0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 1 1 0 0]
d4: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d5: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d6: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 9 6
h0, h1, h2, h3, h4, h5, h6: 2 2 2 2 0 0 0
##################################
n,k 9 7
d0: 2
[9, 7, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 1 0]
[0 0 0 1 0 0 0 1 0]
[0 0 0 0 1 0 0 1 0]
[0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 1 1 0]
d1: 2
[9, 7, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 0]
[0 1 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 1 0]
[0 0 0 1 0 0 0 1 0]
[0 0 0 0 1 0 0 1 0]
[0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 1 1 0]
d2: 1
[9, 7, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 1 0 0]
d3: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d4: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d5: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d6: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 9 7
h0, h1, h2, h3, h4, h5, h6: 2 2 1 0 0 0 0
##################################
n,k 9 8
d0: 2
[9, 8, 2] Cyclic Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 1]
[0 0 1 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 1]
[0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 1 1]
d1: 1
[9, 8, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 1 0]
d2: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d3: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d4: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d5: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d6: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 9 8
h0, h1, h2, h3, h4, h5, h6: 2 1 0 0 0 0 0
##################################
n,k 9 9
d0: 1
[9, 9, 1] Cyclic Linear Code over GF(2)
d1: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d2: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d3: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d4: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d5: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
d6: 0
[9, 0, 9] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 9 9
h0, h1, h2, h3, h4, h5, h6: 1 0 0 0 0 0 0
##################################
n,k 10 1
d0: 9
[10, 1, 9] Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1 1 1 0]
d1: 10
[10, 1, 10] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1 1 1 1]
d2: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d3: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d4: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d5: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d6: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 10 1
h0, h1, h2, h3, h4, h5, h6: 9 10 0 0 0 0 0
##################################
n,k 10 2
d0: 6
[10, 2, 6] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 1 0 0 1 1 0]
[0 1 1 1 1 1 1 0 0 0]
d1: 5
[10, 2, 5] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 1 1 1 0]
[0 1 1 1 1 1 0 0 0 0]
d2: 6
[10, 2, 6] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 0 0 0 1 1 1]
[0 1 1 1 1 1 1 0 0 0]
d3: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d4: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d5: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d6: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 10 2
h0, h1, h2, h3, h4, h5, h6: 6 5 6 0 0 0 0
##################################
n,k 10 3
d0: 5
[10, 3, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 0 1 0 1]
[0 1 0 1 1 0 0 1 1 0]
[0 0 1 1 1 1 1 0 0 0]
d1: 5
[10, 3, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 1 1 0 1 0 1]
[0 1 0 1 1 0 0 1 1 0]
[0 0 1 1 1 1 1 0 0 0]
d2: 4
[10, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 1 1 0 0]
[0 1 0 1 1 0 1 0 0 0]
[0 0 1 1 1 1 0 0 0 0]
d3: 4
[10, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 1 0 0 0]
[0 1 0 1 1 0 1 0 0 0]
[0 0 1 1 1 1 0 0 0 0]
d4: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d5: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d6: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 10 3
h0, h1, h2, h3, h4, h5, h6: 5 5 4 4 0 0 0
##################################
n,k 10 4
d0: 4
[10, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0 1 1 0]
[0 1 0 0 1 0 1 0 1 0]
[0 0 1 0 1 1 0 1 0 0]
[0 0 0 1 1 1 1 0 0 0]
d1: 4
[10, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 1 1 0 1]
[0 1 0 0 1 0 1 0 1 0]
[0 0 1 0 1 1 0 1 0 0]
[0 0 0 1 1 1 1 0 0 0]
d2: 4
[10, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 0 0 1 0]
[0 1 0 0 1 0 1 1 0 0]
[0 0 1 0 1 1 0 1 0 0]
[0 0 0 1 1 1 1 0 0 0]
d3: 4
[10, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 1 0]
[0 1 0 0 1 0 1 1 0 0]
[0 0 1 0 1 1 0 1 0 0]
[0 0 0 1 1 1 1 0 0 0]
d4: 4
[10, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 0 0]
[0 1 0 0 1 0 1 1 0 0]
[0 0 1 0 1 1 0 1 0 0]
[0 0 0 1 1 1 1 0 0 0]
d5: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d6: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 10 4
h0, h1, h2, h3, h4, h5, h6: 4 4 4 4 4 0 0
##################################
n,k 10 5
d0: 3
[10, 5, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0 1 0]
[0 1 0 0 0 1 0 0 1 0]
[0 0 1 0 0 0 1 1 0 0]
[0 0 0 1 0 1 0 1 0 0]
[0 0 0 0 1 1 1 0 0 0]
d1: 4
[10, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 1 0 1]
[0 1 0 0 0 1 1 0 0 1]
[0 0 1 0 0 1 0 1 1 0]
[0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 1 1 1 1 0 0]
d2: 3
[10, 5, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 0 1 0]
[0 1 0 0 0 1 1 1 0 0]
[0 0 1 0 0 0 1 1 0 0]
[0 0 0 1 0 1 0 1 0 0]
[0 0 0 0 1 1 1 0 0 0]
d3: 4
[10, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 0 0 1]
[0 1 0 0 0 0 1 1 1 0]
[0 0 1 0 0 1 0 1 1 0]
[0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 1 1 1 1 0 0]
d4: 4
[10, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 1 1 1]
[0 1 0 0 0 1 0 1 1 1]
[0 0 1 0 0 1 1 0 0 1]
[0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 1 1 1 1 0 0]
d5: 2
[10, 5, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 1 0 0]
[0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 1 1 0 0 0 0]
d6: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 10 5
h0, h1, h2, h3, h4, h5, h6: 3 4 3 4 4 2 0
##################################
n,k 10 6
d0: 3
[10, 6, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 0 1 0 1]
[0 0 1 0 0 0 1 0 0 1]
[0 0 0 1 0 0 0 1 1 0]
[0 0 0 0 1 0 1 0 1 0]
[0 0 0 0 0 1 1 1 0 0]
d1: 3
[10, 6, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 0 1]
[0 1 0 0 0 0 1 0 0 1]
[0 0 1 0 0 0 1 1 1 0]
[0 0 0 1 0 0 0 1 1 0]
[0 0 0 0 1 0 1 0 1 0]
[0 0 0 0 0 1 1 1 0 0]
d2: 3
[10, 6, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 1 1 1]
[0 1 0 0 0 0 0 1 1 1]
[0 0 1 0 0 0 1 0 0 1]
[0 0 0 1 0 0 0 1 1 0]
[0 0 0 0 1 0 1 0 1 0]
[0 0 0 0 0 1 1 1 0 0]
d3: 2
[10, 6, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 0 1 0 1]
[0 0 1 0 0 0 0 1 1 0]
[0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 1 1 0 0 0]
d4: 2
[10, 6, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 1 0 0]
[0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 1 1 0 0 0]
d5: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d6: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 10 6
h0, h1, h2, h3, h4, h5, h6: 3 3 3 2 2 0 0
##################################
n,k 10 7
d0: 2
[10, 7, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 1 0]
[0 1 0 0 0 0 0 1 0 0]
[0 0 1 0 0 0 0 1 0 0]
[0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 1 1 0 0]
d1: 2
[10, 7, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 0 0]
[0 1 0 0 0 0 0 1 0 0]
[0 0 1 0 0 0 0 1 0 0]
[0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 1 1 0 0]
d2: 2
[10, 7, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 0 1 0 1]
[0 0 1 0 0 0 0 1 1 0]
[0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 1 1 0 0]
d3: 2
[10, 7, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 1 0 0]
[0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 1 1 0 0]
d4: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d5: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d6: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 10 7
h0, h1, h2, h3, h4, h5, h6: 2 2 2 2 0 0 0
##################################
n,k 10 8
d0: 2
[10, 8, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 0]
[0 1 0 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 0 1 0]
[0 0 0 1 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 1 0]
[0 0 0 0 0 1 0 0 1 0]
[0 0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 0 1 1 0]
d1: 1
[10, 8, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 0]
[0 1 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 0]
d2: 2
[10, 8, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 0 1 0]
[0 0 0 1 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 1 0]
[0 0 0 0 0 1 0 0 1 0]
[0 0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 0 1 1 0]
d3: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d4: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d5: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d6: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 10 8
h0, h1, h2, h3, h4, h5, h6: 2 1 2 0 0 0 0
##################################
n,k 10 9
d0: 1
[10, 9, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0]
d1: 2
[10, 9, 2] Cyclic Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 1]
[0 0 1 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 1 1]
d2: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d3: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d4: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d5: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d6: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 10 9
h0, h1, h2, h3, h4, h5, h6: 1 2 0 0 0 0 0
##################################
n,k 10 10
d0: 1
[10, 10, 1] Cyclic Linear Code over GF(2)
d1: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d2: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d3: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d4: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d5: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
d6: 0
[10, 0, 10] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 10 10
h0, h1, h2, h3, h4, h5, h6: 1 0 0 0 0 0 0
##################################
n,k 11 1
d0: 11
[11, 1, 11] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1 1 1 1 1]
d1: 10
[11, 1, 10] Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1 1 1 1 0]
d2: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d3: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d4: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d5: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d6: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 11 1
h0, h1, h2, h3, h4, h5, h6: 11 10 0 0 0 0 0
##################################
n,k 11 2
d0: 6
[11, 2, 6] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 1 0 0 1 1 0 0]
[0 1 1 1 1 1 1 0 0 0 0]
d1: 7
[11, 2, 7] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 1 0 0 0 1 1 1]
[0 1 1 1 1 1 1 1 0 0 0]
d2: 6
[11, 2, 6] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 0 0 0 1 1 1 0]
[0 1 1 1 1 1 1 0 0 0 0]
d3: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d4: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d5: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d6: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 11 2
h0, h1, h2, h3, h4, h5, h6: 6 7 6 0 0 0 0
##################################
n,k 11 3
d0: 5
[11, 3, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 0 1 0 1 0]
[0 1 0 1 1 0 0 1 1 0 0]
[0 0 1 1 1 1 1 0 0 0 0]
d1: 6
[11, 3, 6] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 1 0 1 0 1 0 1]
[0 1 0 1 1 1 0 0 1 1 0]
[0 0 1 1 1 1 1 1 0 0 0]
d2: 5
[11, 3, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 0 1 1 0 1]
[0 1 0 1 0 0 0 1 1 1 0]
[0 0 1 1 1 1 1 0 0 0 0]
d3: 4
[11, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 1 0 0 0 0]
[0 1 0 1 1 0 1 0 0 0 0]
[0 0 1 1 1 1 0 0 0 0 0]
d4: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d5: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d6: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 11 3
h0, h1, h2, h3, h4, h5, h6: 5 6 5 4 0 0 0
##################################
n,k 11 4
d0: 4
[11, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0 1 1 0 0]
[0 1 0 0 1 0 1 0 1 0 0]
[0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 1 1 1 1 0 0 0 0]
d1: 5
[11, 4, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 1 0 1 1]
[0 1 0 0 1 0 1 0 1 0 1]
[0 0 1 0 1 1 0 0 1 1 0]
[0 0 0 1 1 1 1 1 0 0 0]
d2: 4
[11, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 0 0 1 0 0]
[0 1 0 0 1 0 1 1 0 0 0]
[0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 1 1 1 1 0 0 0 0]
d3: 4
[11, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 1 0 0]
[0 1 0 0 1 0 1 1 0 0 0]
[0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 1 1 1 1 0 0 0 0]
d4: 4
[11, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 0 0 0]
[0 1 0 0 1 0 1 1 0 0 0]
[0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 1 1 1 1 0 0 0 0]
d5: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d6: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 11 4
h0, h1, h2, h3, h4, h5, h6: 4 5 4 4 4 0 0
##################################
n,k 11 5
d0: 4
[11, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 1 1 0 1]
[0 1 0 0 0 1 0 0 1 1 0]
[0 0 1 0 0 1 0 1 0 1 0]
[0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 1 1 1 1 0 0 0]
d1: 4
[11, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 1 0 1 0]
[0 1 0 0 0 1 1 0 0 1 0]
[0 0 1 0 0 1 0 1 1 0 0]
[0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 1 1 1 1 0 0 0]
d2: 4
[11, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 1 1 0 1]
[0 1 0 0 0 1 1 0 0 1 0]
[0 0 1 0 0 1 0 1 1 0 0]
[0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 1 1 1 1 0 0 0]
d3: 4
[11, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 0 0 1 0]
[0 1 0 0 0 0 1 1 1 0 0]
[0 0 1 0 0 1 0 1 1 0 0]
[0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 1 1 1 1 0 0 0]
d4: 4
[11, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 1 1 0 1]
[0 1 0 0 0 0 1 1 1 1 0]
[0 0 1 0 0 1 0 1 1 0 0]
[0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 1 1 1 1 0 0 0]
d5: 4
[11, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 1 1 1 1]
[0 1 0 0 0 1 1 0 0 0 1]
[0 0 1 0 0 1 1 0 0 1 0]
[0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 1 1 1 1 0 0 0]
d6: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 11 5
h0, h1, h2, h3, h4, h5, h6: 4 4 4 4 4 4 0
##################################
n,k 11 6
d0: 4
[11, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0 0 1 1]
[0 1 0 0 0 0 1 0 1 0 1]
[0 0 1 0 0 0 1 1 0 0 1]
[0 0 0 1 0 0 1 0 1 1 0]
[0 0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 0 1 1 1 1 0 0]
d1: 3
[11, 6, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 0 1 0]
[0 1 0 0 0 0 1 0 0 1 0]
[0 0 1 0 0 0 1 1 1 0 0]
[0 0 0 1 0 0 0 1 1 0 0]
[0 0 0 0 1 0 1 0 1 0 0]
[0 0 0 0 0 1 1 1 0 0 0]
d2: 4
[11, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0 1 0 1]
[0 1 0 0 0 0 1 1 0 0 1]
[0 0 1 0 0 0 0 1 1 1 0]
[0 0 0 1 0 0 1 0 1 1 0]
[0 0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 0 1 1 1 1 0 0]
d3: 3
[11, 6, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0 0 0 1]
[0 1 0 0 0 0 1 0 0 1 0]
[0 0 1 0 0 0 1 1 1 0 0]
[0 0 0 1 0 0 0 1 1 0 0]
[0 0 0 0 1 0 1 0 1 0 0]
[0 0 0 0 0 1 1 1 0 0 0]
d4: 3
[11, 6, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 1 1 1 0 1]
[0 0 1 0 0 0 1 1 1 1 0]
[0 0 0 1 0 0 0 1 1 0 0]
[0 0 0 0 1 0 1 0 1 0 0]
[0 0 0 0 0 1 1 1 0 0 0]
d5: 3
[11, 6, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 1 1 1 1]
[0 1 0 0 0 0 0 1 1 1 1]
[0 0 1 0 0 0 1 0 0 0 1]
[0 0 0 1 0 0 1 0 0 1 0]
[0 0 0 0 1 0 1 0 1 0 0]
[0 0 0 0 0 1 1 1 0 0 0]
d6: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 11 6
h0, h1, h2, h3, h4, h5, h6: 4 3 4 3 3 3 0
##################################
n,k 11 7
d0: 3
[11, 7, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 0 0 1 0 1]
[0 0 1 0 0 0 0 1 0 0 1]
[0 0 0 1 0 0 0 1 1 1 0]
[0 0 0 0 1 0 0 0 1 1 0]
[0 0 0 0 0 1 0 1 0 1 0]
[0 0 0 0 0 0 1 1 1 0 0]
d1: 3
[11, 7, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 1 1]
[0 1 0 0 0 0 0 1 0 1 1]
[0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 1 0 0 0 1 0 0 1]
[0 0 0 0 1 0 0 0 1 1 0]
[0 0 0 0 0 1 0 1 0 1 0]
[0 0 0 0 0 0 1 1 1 0 0]
d2: 3
[11, 7, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 1 0 1]
[0 1 0 0 0 0 0 0 1 0 1]
[0 0 1 0 0 0 0 1 0 0 1]
[0 0 0 1 0 0 0 1 1 1 0]
[0 0 0 0 1 0 0 0 1 1 0]
[0 0 0 0 0 1 0 1 0 1 0]
[0 0 0 0 0 0 1 1 1 0 0]
d3: 2
[11, 7, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1 0]
[0 1 0 0 0 0 0 0 1 0 0]
[0 0 1 0 0 0 0 1 0 0 0]
[0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 1 1 0 0 0]
d4: 2
[11, 7, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 1 1]
[0 1 0 0 0 0 0 0 0 1 1]
[0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 1 1 0 0 0]
d5: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d6: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 11 7
h0, h1, h2, h3, h4, h5, h6: 3 3 3 2 2 0 0
##################################
n,k 11 8
d0: 2
[11, 8, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 0 0]
[0 1 0 0 0 0 0 0 1 0 0]
[0 0 1 0 0 0 0 0 1 0 0]
[0 0 0 1 0 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 1 1 0 0]
d1: 2
[11, 8, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 0 0 1 0 0]
[0 0 1 0 0 0 0 0 1 0 0]
[0 0 0 1 0 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 1 1 0 0]
d2: 2
[11, 8, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1 0]
[0 1 0 0 0 0 0 0 1 0 0]
[0 0 1 0 0 0 0 0 1 0 0]
[0 0 0 1 0 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 1 1 0 0]
d3: 2
[11, 8, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 1 1]
[0 1 0 0 0 0 0 0 0 1 1]
[0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 1 1 0 0]
d4: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d5: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d6: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 11 8
h0, h1, h2, h3, h4, h5, h6: 2 2 2 2 0 0 0
##################################
n,k 11 9
d0: 2
[11, 9, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 0 0 1 0]
[0 0 0 1 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 1 0]
[0 0 0 0 0 1 0 0 0 1 0]
[0 0 0 0 0 0 1 0 0 1 0]
[0 0 0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 0 0 1 1 0]
d1: 2
[11, 9, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1 0]
[0 1 0 0 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 0 0 1 0]
[0 0 0 1 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 1 0]
[0 0 0 0 0 1 0 0 0 1 0]
[0 0 0 0 0 0 1 0 0 1 0]
[0 0 0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 0 0 1 1 0]
d2: 1
[11, 9, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0]
d3: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d4: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d5: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d6: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 11 9
h0, h1, h2, h3, h4, h5, h6: 2 2 1 0 0 0 0
##################################
n,k 11 10
d0: 2
[11, 10, 2] Cyclic Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0 1]
[0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 1 1]
d1: 1
[11, 10, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 1 0]
d2: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d3: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d4: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d5: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d6: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 11 10
h0, h1, h2, h3, h4, h5, h6: 2 1 0 0 0 0 0
##################################
n,k 11 11
d0: 1
[11, 11, 1] Cyclic Linear Code over GF(2)
d1: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d2: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d3: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d4: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d5: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
d6: 0
[11, 0, 11] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 11 11
h0, h1, h2, h3, h4, h5, h6: 1 0 0 0 0 0 0
##################################
n,k 12 1
d0: 11
[12, 1, 11] Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1 1 1 1 1 0]
d1: 12
[12, 1, 12] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1 1 1 1 1 1]
d2: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
d3: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
d4: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
d5: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
d6: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 12 1
h0, h1, h2, h3, h4, h5, h6: 11 12 0 0 0 0 0
##################################
n,k 12 2
d0: 7
[12, 2, 7] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 0 0 0 0 1 1 1 1]
[0 1 1 1 1 1 1 1 0 0 0 0]
d1: 7
[12, 2, 7] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 1 0 0 0 1 1 1 0]
[0 1 1 1 1 1 1 1 0 0 0 0]
d2: 8
[12, 2, 8] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 1 1 0 0 0 1 1 1]
[0 1 1 1 1 1 1 1 1 0 0 0]
d3: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
d4: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
d5: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
d6: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 12 2
h0, h1, h2, h3, h4, h5, h6: 7 7 8 0 0 0 0
##################################
n,k 12 3
d0: 6
[12, 3, 6] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 1 0 1 0 1 0 1 1]
[0 1 0 1 1 1 0 0 1 1 0 0]
[0 0 1 1 1 1 1 1 0 0 0 0]
d1: 6
[12, 3, 6] Line 1 1 1 1 0 0 0 0]
d1: 6
[12, 3, 6] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 1 0 1 0 1 0 1 0]
[0 1 0 1 1 1 0 0 1 1 0 0]
[0 0 1 1 1 1 1 1 0 0 0 0]
d2: 5
[12, 3, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 0 1 1 0 1 0]
[0 1 0 1 0 0 0 1 1 1 0 0]
[0 0 1 1 1 1 1 0 0 0 0 0]
d3: 6
[12, 3, 6] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 0 1 1 0 1]
[0 1 0 1 1 0 0 0 1 1 1 0]
[0 0 1 1 1 1 1 1 0 0 0 0]
d4: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
d5: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
d6: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 12 3
h0, h1, h2, h3, h4, h5, h6: 6 6 5 6 0 0 0
##################################
n,k 12 4
d0: 5
[12, 4, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 0 1 0 0 1]
[0 1 0 0 1 0 1 0 1 0 1 0]
[0 0 1 0 1 1 0 0 1 1 0 0]
[0 0 0 1 1 1 1 1 0 0 0 0]
d1: 5
[12, 4, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 1 0 1 1 0]
[0 1 0 0 1 0 1 0 1 0 1 0]
[0 0 1 0 1 1 0 0 1 1 0 0]
[0 0 0 1 1 1 1 1 0 0 0 0]
d2: 6
[12, 4, 6] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 1 0 1 0 1 1]
[0 1 0 0 1 1 0 1 0 1 0 1]
[0 0 1 0 1 1 1 0 0 1 1 0]
[0 0 0 1 1 1 1 1 1 0 0 0]
d3: 4
[12, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 1 0 0 0]
[0 1 0 0 1 0 1 1 0 0 0 0]
[0 0 1 0 1 1 0 1 0 0 0 0]
[0 0 0 1 1 1 1 0 0 0 0 0]
d4: 4
[12, 4, 4] Quasicyclic of degree 3 Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 0 0 0 0]
[0 1 0 0 1 0 1 1 0 0 0 0]
[0 0 1 0 1 1 0 1 0 0 0 0]
[0 0 0 1 1 1 1 0 0 0 0 0]
d5: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
d6: 0
[12, 0, 12] Cyclic Linear Code over GF(2)
***Summary of hulls from 0 to 6:***
n,k 12 4
h0, h1, h2, h3, h4, h5, h6: 5 5 6 4 4 0 0
Part 2
Calculation based on building-up construction fro n >= 12 and k>=5.
There are given two generator matrices. The first one is a construction form(I, II, III, IV).
The second one is a row echelon form of the construction form.
"h=1"
n=12
k=1
[12,1,12]
Repetition code
k=2
[12, 2, 7] Exhaustive search
k=3
[12, 3, 6] code. See above or Magma code
> C:=BKLC(GF(2), 12, 3);
> C;
[12, 3, 6] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 0 1 1 0 1 0]
[0 1 0 1 1 0 0 1 0 1 1 0]
[0 0 1 1 1 1 0 0 1 1 1 1]
> Hull(C);
[12, 1, 8] Quasicyclic of degree 6 Linear Code over GF(2)
Generator matrix:
[0 0 1 1 1 1 0 0 1 1 1 1]
k=4
[12, 4, 5] Exhaustive search (see above or below)
[12, 4, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 1 0 1 1 0]
[0 1 0 0 1 0 1 0 1 0 1 0]
[0 0 1 0 1 1 0 0 1 1 0 0]
[0 0 0 1 1 1 1 1 0 0 0 0]
k=5
[12, 5, 4] code from Magma result
> C:=BKLC(GF(2), 12, 5);
> C;
[12, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 0 0 0 0 1 1 0]
[0 1 0 1 0 0 0 0 0 1 0 1]
[0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 1 0 1 1 0]
[0 0 0 0 0 1 0 1 0 1 0 1]
> Hull(C);
[12, 1, 4] Quasicyclic of degree 3 Linear Code over GF(2)
Generator matrix:
[1 1 1 1 0 0 0 0 0 0 0 0]
k=6 %%%
Construction I from a [10, 5, 3] code with h=0
88
[1 0 1 1 1 1 0 1 1 1 0 0]
[1 1 1 0 0 0 1 0 0 0 1 0]
[1 1 0 1 0 0 1 0 0 0 0 1]
[1 1 0 0 1 0 0 0 0 0 1 1]
[0 0 0 0 0 1 1 0 0 1 0 1]
[0 0 0 0 0 0 0 0 1 1 1 1]
[12, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 1 1 0 1 0 1]
[0 1 0 0 0 0 1 1 0 1 1 0]
[0 0 1 0 1 0 1 0 0 0 0 1]
[0 0 0 1 1 0 1 0 0 0 1 0]
[0 0 0 0 0 1 1 0 0 1 0 1]
[0 0 0 0 0 0 0 0 1 1 1 1]
hull dim 1
[ <0, 1>, <4, 12>, <5, 14>, <6, 12>, <7, 12>, <8, 7>, <9, 6> ]
k=7
Construction III from a [10, 6, 3] code with h=0
38 %%%
[1 1 1 1 1 1 0 1 1 0 0 0]
[1 0 1 0 0 0 1 0 0 1 0 0]
[1 0 0 1 0 0 1 0 0 0 1 0]
[1 0 0 0 1 0 0 0 0 1 1 0]
[1 0 0 0 0 1 1 0 0 0 0 1]
[1 0 0 0 0 0 0 1 0 1 0 1]
[1 0 0 0 0 0 0 0 1 0 1 1]
[12, 7, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 0 1 1]
[0 1 0 0 0 0 1 0 1 1 0 0]
[0 0 1 0 0 0 1 0 1 1 1 1]
[0 0 0 1 0 0 1 0 1 0 0 1]
[0 0 0 0 1 0 0 0 1 1 0 1]
[0 0 0 0 0 1 1 0 1 0 1 0]
[0 0 0 0 0 0 0 1 1 1 1 0]
hull dim 1
[ <0, 1>, <4, 38>, <6, 52>, <8, 33>, <10, 4> ]
k=8
Construction III from a [10, 7, 2] code with h=0
262 %%%
[1 1 1 1 1 1 0 0 0 0 1 1]
[1 0 1 0 0 0 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 1 0 0 0 0 1 1 0]
[1 0 0 0 0 1 0 0 0 1 1 1]
[0 0 0 0 0 0 1 0 0 0 1 1]
[1 0 0 0 0 0 0 1 0 1 0 1]
[0 0 0 0 0 0 0 0 1 1 1 1]
[12, 8, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 0 1 0 1]
[0 1 0 0 0 0 0 1 0 0 0 1]
[0 0 1 0 0 0 0 1 0 1 1 0]
[0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 1 0 1 0 0 1 0]
[0 0 0 0 0 0 1 0 0 0 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1]
hull dim 1
[ <0, 1>, <3, 17>, <4, 38>, <5, 44>, <6, 52>, <7, 54>, <8, 33>, <9, 12>, <10,
4>, <11, 1> ]
k=9
Construction I from a [10, 8, 2] code with h=0
1 %%%
[1 0 1 0 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 1 1]
[12, 9, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 1 1]
hull dim 1
[ <0, 1>, <2, 37>, <4, 162>, <6, 210>, <8, 93>, <10, 9> ]
k=10
[12, 10, 1] code, exhaustive search and direct computation
[12, 10, 1] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 1]
> Hull(C);
[12, 1, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 0 0 0 0 0]
k=11
[12, 11, 2] code. Magma data
[12, 11, 2] Cyclic Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0 0 1]
[0 0 1 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 1 1]
> Hull(C);
[12, 1, 12] Cyclic Linear Code over GF(2)
Generator matrix:
[1 1 1 1 1 1 1 1 1 1 1 1]
n=13
k=1
[13, 1, 12] code.
Self-orthogonal [13, 1, 12] code
Generator matrix
[1 1 1 1 1 1 1 1 1 1 1 1 0]
k=2
Construction I
> C:=BKLC(GF(2), 13, 2);
> C;
[13, 2, 8] Linear Code over GF(2)
Generator matrix:
[1 1 1 1 0 0 0 0 0 1 1 1 1]
[0 0 0 0 1 1 1 1 1 1 1 1 1]
> Hull(C);
[13, 1, 8] Linear Code over GF(2)
Generator matrix:
[1 1 1 1 0 0 0 0 0 1 1 1 1]
k=3
Construction I from a [11, 2, 6] code with h=0
133 %%%
[1 0 1 0 1 1 0 0 0 1 1 0 0]
[1 1 1 1 1 0 0 0 1 1 0 0 1]
[0 0 0 0 0 1 1 1 1 1 0 0 1]
[13, 3, 6] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 1 0 1 1 1 0 1 0 1]
[0 1 0 1 0 0 1 1 0 1 1 0 0]
[0 0 0 0 0 1 1 1 1 1 0 0 1]
hull dim 1
[ <0, 1>, <6, 4>, <8, 3> ]
k=4
Construction I from a [11, 3, 5] code with h=0
165 %%%
[1 0 0 1 1 1 0 1 1 1 1 0 0]
[1 1 1 0 0 1 1 0 1 1 0 1 1]
[0 0 0 1 0 1 1 1 0 1 0 0 1]
[0 0 0 0 1 1 1 1 1 0 0 1 0]
[13, 4, 6] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 1 0 0 1 1 1]
[0 1 1 0 0 0 1 1 1 1 1 0 0]
[0 0 0 1 0 1 1 1 0 1 0 0 1]
[0 0 0 0 1 1 1 1 1 0 0 1 0]
hull dim 1
[ <0, 1>, <6, 6>, <7, 6>, <8, 1>, <9, 2> ]
k=5
Construction I from a [11, 4, 4] code with h=0
83 %%%
[1 0 1 1 1 0 1 1 1 1 0 0 0]
[1 1 1 0 0 0 1 1 0 0 1 0 0]
[1 1 0 1 0 0 1 0 1 0 0 1 0]
[1 1 0 0 1 0 1 0 1 0 1 0 1]
[0 0 0 0 0 1 1 1 0 0 0 1 1]
[13, 5, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 1 0 0 1 1 1 0]
[0 1 0 0 0 0 0 0 1 1 0 1 1]
[0 0 1 0 1 0 0 1 1 0 0 0 1]
[0 0 0 1 1 0 0 0 0 0 1 1 1]
[0 0 0 0 0 1 1 1 0 0 0 1 1]
hull dim 1
[ <0, 1>, <5, 5>, <6, 12>, <7, 7>, <8, 3>, <9, 3>, <11, 1> ]
k=6
Construction I from a [11, 5, 4] code with h=0
9 %%%
[1 0 1 0 0 1 1 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 1 0 1 1 1]
[0 0 0 1 0 0 0 1 0 1 0 1 0]
[0 0 0 0 1 0 0 1 1 0 0 1 1]
[1 1 0 0 0 1 0 0 0 1 1 1 0]
[1 1 0 0 0 0 1 1 1 0 1 0 1]
[13, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0 1 1 0 0 1]
[0 1 0 0 0 0 0 1 0 1 1 0 0]
[0 0 1 0 0 0 1 1 0 0 0 1 0]
[0 0 0 1 0 0 0 1 0 1 0 1 0]
[0 0 0 0 1 0 0 1 1 0 0 1 1]
[0 0 0 0 0 1 1 1 1 1 0 1 1]
hull dim 1
[ <0, 1>, <4, 12>, <5, 8>, <6, 12>, <7, 10>, <8, 7>, <9, 12>, <11, 2> ]
k=7
Construction I from a [11, 6, 4] code with h=0
9 %%%
[1 0 1 0 0 1 1 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 1 0 1 1]
[0 0 0 1 0 0 0 1 0 1 1 1 1]
[0 0 0 0 1 0 0 1 0 1 0 0 1]
[1 1 0 0 0 1 0 0 0 1 1 0 1]
[1 1 0 0 0 0 1 1 0 1 0 1 0]
[0 0 0 0 0 0 0 0 1 1 1 1 0]
[13, 7, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0 0 0 1 1 0]
[0 1 0 0 0 0 0 1 0 1 1 0 0]
[0 0 1 0 0 0 1 1 0 0 0 0 1]
[0 0 0 1 0 0 0 1 0 1 1 1 1]
[0 0 0 0 1 0 0 1 0 1 0 0 1]
[0 0 0 0 0 1 1 1 0 0 1 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1 0]
hull dim 1
[ <0, 1>, <4, 26>, <6, 48>, <8, 45>, <10, 8> ]
k=8
Construction I from a [11, 7, 3] code with h=0
8 %%%
[1 0 1 0 0 1 1 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 1 0 1 1]
[0 0 0 1 0 0 0 0 0 1 1 0 1]
[0 0 0 0 1 0 0 0 0 0 1 1 0]
[1 1 0 0 0 1 0 0 0 1 1 1 0]
[1 1 0 0 0 0 1 0 0 1 1 1 1]
[0 0 0 0 0 0 0 1 0 0 1 0 1]
[0 0 0 0 0 0 0 0 1 0 0 1 1]
[13, 8, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0 0 0 1 0 1]
[0 1 0 0 0 0 0 0 0 1 0 1 0]
[0 0 1 0 0 0 1 0 0 0 1 0 0]
[0 0 0 1 0 0 0 0 0 1 1 0 1]
[0 0 0 0 1 0 0 0 0 0 1 1 0]
[0 0 0 0 0 1 1 0 0 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 1 0 1]
[0 0 0 0 0 0 0 0 1 0 0 1 1]
hull dim 1
[ <0, 1>, <3, 12>, <4, 25>, <5, 34>, <6, 50>, <7, 58>, <8, 45>, <9, 22>, <10,
6>, <11, 2>, <12, 1> ]
k=9
Construction I from a [11, 8, 2] code with h=0
1 %%%
[1 0 1 0 0 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 1 1]
[13, 9, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 1 1]
hull dim 1
[ <0, 1>, <2, 37>, <4, 162>, <6, 210>, <8, 93>, <10, 9> ]
k=10
Construction I from a [11, 9, 2] code with h=0
1 %%%
[1 0 1 0 0 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 0 1 0]
[0 0 0 1 0 0 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 0 0 1 1]
[0 0 0 0 0 1 0 0 0 0 0 1 0]
[0 0 0 0 0 0 1 0 0 0 0 1 0]
[0 0 0 0 0 0 0 1 0 0 0 1 0]
[0 0 0 0 0 0 0 0 1 0 0 1 0]
[0 0 0 0 0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 1 1 0]
[13, 10, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 1 0 0 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 0 0 1 1]
[0 0 0 0 0 1 0 0 0 0 0 1 0]
[0 0 0 0 0 0 1 0 0 0 0 1 0]
[0 0 0 0 0 0 0 1 0 0 0 1 0]
[0 0 0 0 0 0 0 0 1 0 0 1 0]
[0 0 0 0 0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 1 1 0]
hull dim 1
[ <0, 1>, <2, 37>, <3, 9>, <4, 162>, <5, 93>, <6, 210>, <7, 210>, <8, 93>, <9,
162>, <10, 9>, <11, 37>, <13, 1> ]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
"h=2"
n=12
k=2
[12, 2, 8] self-orthogonal code
d2: 8
[12, 2, 8] Linear Code over GF(2)
Generator matrix:
[1 0 1 1 1 1 0 0 0 1 1 1]
[0 1 1 1 1 1 1 1 1 0 0 0]
k=3
[12, 3, 5] exhaustive method (see above or below)
d2: 5
[12, 3, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 0 1 1 0 1 0]
[0 1 0 1 0 0 0 1 1 1 0 0]
[0 0 1 1 1 1 1 0 0 0 0 0]
k=4
[12, 4, 6] exhaustive method (see above or below)
d2: 6
[12, 4, 6] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 1 0 1 0 1 1]
[0 1 0 0 1 1 0 1 0 1 0 1]
[0 0 1 0 1 1 1 0 0 1 1 0]
[0 0 0 1 1 1 1 1 1 0 0 0]
k=5
Construction III from a [10, 4, 4] code with h=0
9 %%%
[1 1 0 0 0 1 1 0 0 0 0 0]
[1 0 1 0 0 0 1 1 1 0 0 1]
[0 0 0 1 0 0 0 1 1 1 0 1]
[1 0 0 0 1 0 1 0 1 0 1 1]
[1 0 0 0 0 1 0 0 1 1 1 1]
[12, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 0 1 1 1 1]
[0 1 0 0 0 0 1 0 1 1 1 1]
[0 0 1 0 0 1 1 1 0 1 1 0]
[0 0 0 1 0 0 0 1 1 1 0 1]
[0 0 0 0 1 1 1 0 0 1 0 0]
hull dim 2
[ <0, 1>, <4, 4>, <5, 7>, <6, 8>, <7, 7>, <8, 3>, <9, 1>, <11, 1> ]
k=6
[12, 6, 4]
Construction III from a [10, 5, 3] code with h=0
70 %%%
[1 1 1 1 1 1 0 0 1 1 0 0]
[1 0 1 0 0 0 1 0 0 0 1 0]
[1 0 0 1 0 0 1 0 0 0 0 1]
[1 0 0 0 1 0 0 0 0 0 1 1]
[0 0 0 0 0 1 1 0 0 1 0 1]
[0 0 0 0 0 0 0 0 1 1 1 1]
[12, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0 0 0 0 1 1]
[0 1 0 0 0 0 1 0 0 1 1 0]
[0 0 1 0 1 0 1 0 0 0 0 1]
[0 0 0 1 1 0 1 0 0 0 1 0]
[0 0 0 0 0 1 1 0 0 1 0 1]
[0 0 0 0 0 0 0 0 1 1 1 1]
hull dim 2
[ <0, 1>, <4, 26>, <6, 24>, <8, 13> ]
k=7
Construction III from a [10, 6, 3] code with h=0
10 %%%
[1 1 1 1 0 1 1 0 0 0 0 0]
[1 0 1 0 0 0 0 0 1 0 1 1]
[1 0 0 1 0 0 0 0 1 1 0 1]
[0 0 0 0 1 0 0 0 0 1 1 0]
[1 0 0 0 0 1 0 0 0 1 0 1]
[1 0 0 0 0 0 1 0 1 1 1 1]
[0 0 0 0 0 0 0 1 1 1 1 0]
[12, 7, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 0 1 1 1 1]
[0 1 0 0 0 0 1 0 0 0 1 1]
[0 0 1 0 0 0 1 0 0 1 0 0]
[0 0 0 1 0 0 1 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 1 1 0 1 0 1 0]
[0 0 0 0 0 0 0 1 1 1 1 0]
hull dim 2
[ <0, 1>, <3, 8>, <4, 16>, <5, 24>, <6, 30>, <7, 24>, <8, 15>, <9, 8>, <10, 2> ]
k=8
Construction III from a [10, 7, 2] code with h=0
134 %%%
[1 1 1 1 1 1 0 0 0 1 1 0]
[0 0 1 0 0 0 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 0 1 0 1]
[1 0 0 0 1 0 0 0 0 1 1 0]
[1 0 0 0 0 1 0 0 0 1 1 1]
[1 0 0 0 0 0 1 0 0 0 1 1]
[1 0 0 0 0 0 0 1 0 1 0 1]
[0 0 0 0 0 0 0 0 1 1 1 1]
[12, 8, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 0 1 0 1]
[0 1 0 0 0 0 0 1 0 1 0 0]
[0 0 1 0 0 0 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 1 0 0 1 0 0 1 1]
[0 0 0 0 0 1 0 1 0 0 1 0]
[0 0 0 0 0 0 1 1 0 1 1 0]
[0 0 0 0 0 0 0 0 1 1 1 1]
hull dim 2
[ <0, 1>, <3, 16>, <4, 39>, <5, 48>, <6, 48>, <7, 48>, <8, 39>, <9, 16>, <12, 1>
]
k=9
Construction III from a [10, 8, 2] code with h=0
3
[1 1 0 1 1 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 1 1]
[1 0 0 1 0 0 0 0 0 0 1 1]
[1 0 0 0 1 0 0 0 0 0 1 0]
[0 0 0 0 0 1 0 0 0 0 1 1]
[0 0 0 0 0 0 1 0 0 0 1 1]
[0 0 0 0 0 0 0 1 0 0 1 1]
[0 0 0 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 1 1 1]
[12, 9, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0 0 0 0 1 0]
[0 1 0 0 1 0 0 0 0 0 1 1]
[0 0 1 0 0 0 0 0 0 0 1 1]
[0 0 0 1 1 0 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 1 1]
[0 0 0 0 0 0 1 0 0 0 1 1]
[0 0 0 0 0 0 0 1 0 0 1 1]
[0 0 0 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 1 1 1]
hull dim 2
[ <0, 1>, <2, 15>, <3, 22>, <4, 42>, <5, 120>, <6, 126>, <7, 84>, <8, 69>, <9,
24>, <10, 3>, <11, 6> ]
k=10
Construction III from a [10, 9, 1] code with h=0
1 %%%
[1 1 1 1 1 1 1 1 1 1 1 1]
[1 0 1 0 0 0 0 0 0 0 0 0]
[1 0 0 0 1 0 0 0 0 0 0 0]
[1 0 0 0 0 1 0 0 0 0 0 0]
[1 0 0 0 0 0 1 0 0 0 0 0]
[1 0 0 0 0 0 0 1 0 0 0 0]
[1 0 0 0 0 0 0 0 1 0 0 0]
[1 0 0 0 0 0 0 0 0 1 0 0]
[1 0 0 0 0 0 0 0 0 0 1 0]
[1 0 0 0 0 0 0 0 0 0 0 1]
[12, 10, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 1]
[0 1 0 1 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 1 1]
hull dim 2
[ <0, 1>, <2, 46>, <4, 255>, <6, 420>, <8, 255>, <10, 46>, <12, 1> ]
n=13
k=2
Direct calculation or
There exists a self-orthogonal [13,2 8] code
[13, 2, 8] code
Generator matrix
[1 1 1 1 1 1 1 1 0 0 0 0 0]
[0 0 0 0 1 1 1 1 1 1 1 1 0]
k=3
Construction III from a [11, 2, 6] code with h=0
130 %%%
[1 1 1 1 0 0 0 1 1 0 1 1 0]
[0 0 1 1 1 0 0 0 1 1 1 0 1]
[1 0 0 0 0 1 1 1 1 1 1 0 1]
[13, 3, 7] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 1 1 1 0 1]
[0 1 0 0 1 1 1 0 1 0 1 1 0]
[0 0 1 1 1 0 0 0 1 1 1 0 1]
hull dim 2
[ <0, 1>, <7, 4>, <8, 3> ]
k=4
Construction III from a [11, 3, 5] code with h=0
46 %%%
[1 1 0 1 1 1 0 1 1 1 0 0 0]
[1 0 1 0 0 1 1 0 1 1 0 1 1]
[0 0 0 1 0 1 1 1 0 1 0 0 1]
[0 0 0 0 1 1 1 1 1 0 0 1 0]
[13, 4, 6] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 1 1 0 1 1 0 1 1]
[0 1 1 0 0 0 1 1 1 1 0 0 0]
[0 0 0 1 0 1 1 1 0 1 0 0 1]
[0 0 0 0 1 1 1 1 1 0 0 1 0]
hull dim 2
[ <0, 1>, <6, 12>, <8, 3> ]
k=5
Construction III from a [11, 4, 4] code with h=0
37 %%%
[1 1 0 1 1 1 1 0 1 1 0 0 0]
[1 0 1 0 0 0 1 1 0 0 1 0 0]
[1 0 0 1 0 0 1 0 1 0 0 1 0]
[1 0 0 0 1 0 1 0 1 0 1 0 1]
[0 0 0 0 0 1 1 1 0 0 0 1 1]
[13, 5, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 1 0 1 0 1 0 1]
[0 1 0 0 1 0 1 1 0 1 0 0 1]
[0 0 1 0 1 0 0 1 1 0 0 0 1]
[0 0 0 1 1 0 0 0 0 0 1 1 1]
[0 0 0 0 0 1 1 1 0 0 0 1 1]
hull dim 2
[ <0, 1>, <5, 8>, <6, 10>, <7, 4>, <8, 3>, <9, 4>, <10, 2> ]
k=6
Construction III from a [11, 5, 4] code with h=0
1 %%%
[1 1 1 1 0 0 0 0 0 0 0 0 0]
[1 0 1 0 0 0 0 0 1 0 1 1 1]
[1 0 0 1 0 0 0 1 0 1 0 1 0]
[0 0 0 0 1 0 0 1 1 0 0 1 1]
[0 0 0 0 0 1 0 0 0 1 1 1 0]
[0 0 0 0 0 0 1 1 1 0 1 0 1]
[13, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 0 0 1 0 1 0 1 0]
[0 1 0 1 0 0 0 0 1 0 1 1 1]
[0 0 1 1 0 0 0 1 1 1 1 0 1]
[0 0 0 0 1 0 0 1 1 0 0 1 1]
[0 0 0 0 0 1 0 0 0 1 1 1 0]
[0 0 0 0 0 0 1 1 1 0 1 0 1]
hull dim 2
[ <0, 1>, <4, 8>, <5, 10>, <6, 14>, <7, 14>, <8, 7>, <9, 6>, <10, 2>, <11, 2> ]
k=7
Construction I from a [11, 6, 3] code with h=1
d1: 3
[11, 6, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 0 1 0]
[0 1 0 0 0 0 1 0 0 1 0]
[0 0 1 0 0 0 1 1 1 0 0]
[0 0 0 1 0 0 0 1 1 0 0]
[0 0 0 0 1 0 1 0 1 0 0]
[0 0 0 0 0 1 1 1 0 0 0]
575 %%%
[1 0 1 1 0 0 0 1 1 1 0 1 1]
[1 1 1 0 0 0 0 0 0 1 0 1 0]
[1 1 0 1 0 0 0 0 1 0 0 1 0]
[0 0 0 0 1 0 0 0 1 1 1 0 0]
[1 1 0 0 0 1 0 0 0 1 1 0 0]
[1 1 0 0 0 0 1 0 1 0 1 0 0]
[1 1 0 0 0 0 0 1 1 1 0 0 0]
[13, 7, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 0 0 0 1 1]
[0 1 0 0 0 0 0 0 1 1 0 1 1]
[0 0 1 0 0 0 0 1 1 0 0 1 0]
[0 0 0 1 0 0 0 1 0 1 0 1 0]
[0 0 0 0 1 0 0 0 1 1 1 0 0]
[0 0 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 1 1 0 1 1 0 0]
hull dim 2
[ <0, 1>, <4, 19>, <5, 20>, <6, 24>, <7, 24>, <8, 19>, <9, 20>, <12, 1> ]
k=8
Construction III from a [11, 7, 3] code with h=0
134 %%%
[1 1 1 1 1 0 1 0 0 0 1 1 0]
[0 0 1 0 0 0 0 0 0 1 0 1 1]
[0 0 0 1 0 0 0 0 0 1 1 0 1]
[1 0 0 0 1 0 0 0 0 0 1 1 0]
[0 0 0 0 0 1 0 0 0 1 1 1 0]
[1 0 0 0 0 0 1 0 0 1 1 1 1]
[1 0 0 0 0 0 0 1 0 0 1 0 1]
[1 0 0 0 0 0 0 0 1 0 0 1 1]
[13, 8, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 0 0 1 1]
[0 1 0 0 0 0 0 0 1 1 0 1 0]
[0 0 1 0 0 0 0 0 0 1 0 1 1]
[0 0 0 1 0 0 0 0 0 1 1 0 1]
[0 0 0 0 1 0 0 0 1 0 1 0 1]
[0 0 0 0 0 1 0 0 0 1 1 1 0]
[0 0 0 0 0 0 1 0 1 1 1 0 0]
[0 0 0 0 0 0 0 1 1 0 1 1 0]
hull dim 2
[ <0, 1>, <4, 55>, <6, 96>, <8, 87>, <10, 16>, <12, 1> ]
k=9
Magma BKLC(GF(2), 13, 9)
> C:=BKLC(GF(2), 13, 9);
> C;
[13, 9, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 0 0 1 0 1 0 0]
[0 0 1 0 0 0 0 0 1 0 1 1 1]
[0 0 0 1 0 0 0 0 1 0 0 0 1]
[0 0 0 0 1 0 0 0 1 0 0 1 0]
[0 0 0 0 0 1 0 0 0 0 1 0 1]
[0 0 0 0 0 0 1 0 0 0 1 1 0]
[0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1 1 1 1]
> Hull(C);
[13, 2, 8] Linear Code over GF(2)
Generator matrix:
[0 1 1 0 0 1 1 1 1 0 0 1 1]
[0 0 0 1 1 1 1 1 1 1 1 0 0]
k=10
[13, 10, 2] direct method
> C:=LinearCode
> [1,0,0,0,0,0,0,0,0,0,1,0,0],
> [0,1,0,0,0,0,0,0,0,0,0,1,0],
> [0,0,1,0,0,0,0,0,0,0,0,0,1],
> [0,0,0,1,0,0,0,0,0,0,0,0,1],
> [0,0,0,0,1,0,0,0,0,0,0,0,1],
> [0,0,0,0,0,1,0,0,0,0,0,0,1],
> [0,0,0,0,0,0,1,0,0,0,0,0,1],
> [0,0,0,0,0,0,0,1,0,0,0,0,1],
> [0,0,0,0,0,0,0,0,1,0,0,0,1],
> [0,0,0,0,0,0,0,0,0,1,0,0,1]>;
> C;
[13, 10, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 1 0 0]
[0 1 0 0 0 0 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 0 1]
> Hull(C);
[13, 2, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 1 0 0]
[0 1 0 0 0 0 0 0 0 0 0 1 0]
k=11
[13, 11, 1] direct method
> C:=LinearCode
> [1,0,0,0,0,0,0,0,0,0,0,1,0],
> [0,1,0,0,0,0,0,0,0,0,0,0,1],
> [0,0,1,0,0,0,0,0,0,0,0,0,0],
> [0,0,0,1,0,0,0,0,0,0,0,0,0],
> [0,0,0,0,1,0,0,0,0,0,0,0,0],
> [0,0,0,0,0,1,0,0,0,0,0,0,0],
> [0,0,0,0,0,0,1,0,0,0,0,0,0],
> [0,0,0,0,0,0,0,1,0,0,0,0,0],
> [0,0,0,0,0,0,0,0,1,0,0,0,0],
> [0,0,0,0,0,0,0,0,0,1,0,0,0],
> [0,0,0,0,0,0,0,0,0,0,1,0,0]>;
> C;
[13, 11, 1] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 1 0]
[0 1 0 0 0 0 0 0 0 0 0 0 1]
[0 0 1 0 0 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0]
> Hull(C);
[13, 2, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 1 0]
[0 1 0 0 0 0 0 0 0 0 0 0 1]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
"h=3"
n=12 (h=3)
k=3
Exhaustive search or fact that
there exists a self-orthogonal [12, 3, 6] code
d3: 6
[12, 3, 6] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 0 1 1 0 1]
[0 1 0 1 1 0 0 0 1 1 1 0]
[0 0 1 1 1 1 1 1 0 0 0 0]
k=4
d3: 4
[12, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 1 0 0 0]
[0 1 0 0 1 0 1 1 0 0 0 0]
[0 0 1 0 1 1 0 1 0 0 0 0]
[0 0 0 1 1 1 1 0 0 0 0 0]
k=5
Consruction III from a [10, 4, 4] code with h=2
2 %%%
C1:=LinearCode
[1, 1, 1, 1, 0, 0, 0 ,0 ,0 ,0 ,0 ,0],
[1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0],
[1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0],
[0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0]>;
C2: [12, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 0 1 0 1 1 0 0]
[0 1 0 1 0 0 0 1 1 1 1 0]
[0 0 1 1 0 0 1 1 0 0 1 0]
[0 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0]
hull dim 3
[ <0, 1>, <4, 4>, <5, 14>, <6, 8>, <8, 3>, <9, 2> ]
C1 eq C2; true
k=6
Ground gen. matrix:
[1 0 0 0 0 1 1 0 0 1]
[0 1 0 0 0 0 1 1 1 0]
[0 0 1 0 0 1 0 1 1 0]
[0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 1 1 1 1 0 0]
Construction III from a [10, 5, 3] code with h=2
2 %%%
(1 1 0 0 0 0 0 0 0 0)
C:=LinearCode<GF(2), 12 |
[1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1],
[1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0],
[0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0]>;
C;
[12, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 0 0 0 1 1 1 0]
[0 1 0 1 0 0 0 1 1 0 0 1]
[0 0 1 1 0 0 0 1 0 1 1 1]
[0 0 0 0 1 0 0 1 0 1 1 0]
[0 0 0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 0 0 1 1 1 1 0 0]
hull dim 3
[ <0, 1>, <4, 10>, <5, 20>, <6, 8>, <7, 8>, <8, 13>, <9, 4> ]
k=7
Construction III from a [10, 6, 3] code with h=1
(Comment: A [12, 7, 4] Linear Code over GF(2) with h=3 is unique)
d1: 3
[10, 6, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 0 1]
[0 1 0 0 0 0 1 0 0 1]
[0 0 1 0 0 0 1 1 1 0]
[0 0 0 1 0 0 0 1 1 0]
[0 0 0 0 1 0 1 0 1 0]
[0 0 0 0 0 1 1 1 0 0]
342 %%%
[1 1 1 1 1 1 1 1 1 1 1 1]
[1 0 1 0 0 0 0 0 0 1 0 1]
[1 0 0 1 0 0 0 0 1 0 0 1]
[0 0 0 0 1 0 0 0 1 1 1 0]
[1 0 0 0 0 1 0 0 0 1 1 0]
[1 0 0 0 0 0 1 0 1 0 1 0]
[1 0 0 0 0 0 0 1 1 1 0 0]
[12, 7, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 1 1 0 0]
[0 1 0 0 0 0 0 0 1 1 0 1]
[0 0 1 0 0 0 0 1 1 0 0 1]
[0 0 0 1 0 0 0 1 0 1 0 1]
[0 0 0 0 1 0 0 0 1 1 1 0]
[0 0 0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 0 0 1 1 0 1 1 0]
hull dim 3
[ <0, 1>, <4, 39>, <6, 48>, <8, 39>, <12, 1> ]
k=8
Construction III from a [10, 7, 2] code with h=2
Ground gen. matrix:
[1 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 1 0 0]
[0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 1 1 0 0]
(1 1 0 0 0 0 0 0 0 0) % added vector
C:=LinearCode
[1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1],
[1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0]>;
> C;
[12, 8, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 0 0 0 0 0 1 0]
[0 1 0 1 0 0 0 0 0 0 0 1]
[0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0]
> Hull(C);
[12, 3, 4] Linear Code over GF(2)
Generator matrix:
[1 1 0 0 0 0 0 0 0 0 1 1]
[0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 1 1 1 1 1 1 0 0]
> WeightDistribution(C);
[ <0, 1>, <2, 15>, <3, 4>, <4, 18>, <5, 60>, <6, 46>, <7, 60>, <8, 45>, <9, 4>,
<10, 3> ]
k=9
Construciton II from a [10, 8, 2] code with h=2
Ground gen. matrix:
[1 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 1 0]
[0 0 1 0 0 0 0 0 1 0]
[0 0 0 1 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 1 0]
[0 0 0 0 0 1 0 0 1 0]
[0 0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 0 1 1 0]
1 %%%
(0 0 0 0 0 0 0 0 0 0) %% added vector
C:=LinearCode
[1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0]>;
[12, 9, 2] Linear Code over GF(2)
Generator matrix:
[1 1 0 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 0 1 0]
[0 0 0 0 0 1 0 0 0 0 1 0]
[0 0 0 0 0 0 1 0 0 0 1 0]
[0 0 0 0 0 0 0 1 0 0 1 0]
[0 0 0 0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 0 0 0 1 1 0]
hull dim 3
[ <0, 1>, <2, 30>, <4, 127>, <6, 196>, <8, 127>, <10, 30>, <12, 1> ]
n=13 (h=3)
k=3
There exists a binary self-orthogonal [13, 3, 6] code,
In fact such a code is unique by Bouyukliev, Bouyuklieva, Gulliver.
"Classification of Optimal Binary Self-Orthogonal Codes"
Below is an example of a self-orthogonal [13, 3, 6] code.
Generator matrix:
[1 1 1 0 0 1 0 0 0 1 1 1 1]
[0 0 0 1 0 1 1 1 1 0 1 1 1]
[0 0 0 0 1 1 1 1 1 1 0 0 0]
> Hull(C);
[13, 3, 6] Linear Code over GF(2)
Generator matrix:
[1 1 1 0 0 1 0 0 0 1 1 1 1]
[0 0 0 1 0 1 1 1 1 0 1 1 1]
[0 0 0 0 1 1 1 1 1 1 0 0 0]
> WeightDistribution(C);
[ <0, 1>, <6, 3>, <8, 3>, <10, 1> ]
k=4
Construction I from a [11, 3, 5] code with h=2
11 %%%
[1 0 1 1 1 1 1 0 0 0 0 0 0]
[0 0 1 0 0 0 1 1 0 1 1 0 1]
[0 0 0 1 0 1 0 0 0 1 1 1 0]
[1 1 0 0 1 1 1 1 1 0 0 0 0]
[13, 4, 5] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0 1 0 0 0 1 1]
[0 1 0 0 0 1 1 0 1 0 0 1 1]
[0 0 1 0 0 0 1 1 0 1 1 0 1]
[0 0 0 1 0 1 0 0 0 1 1 1 0]
hull dim 3
[ <0, 1>, <5, 3>, <6, 3>, <7, 4>, <8, 3>, <9, 1>, <10, 1> ]
k=5
Construction I froma [11, 4, 4] code with h=2
3 %%%
[1 0 1 1 1 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 1 1 0 0 1 0 0]
[1 1 0 1 0 0 1 0 1 1 0 0 0]
[1 1 0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0 0]
[13, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0 1 1 1 1 0 0]
[0 1 0 0 0 0 1 0 1 0 1 0 0]
[0 0 1 0 1 0 0 0 0 1 1 0 0]
[0 0 0 1 1 0 0 1 1 0 0 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0 0]
hull dim 3
[ <0, 1>, <4, 10>, <6, 16>, <8, 5> ]
Construction IV from a [11, 4, 4] code with h=3
250 %%%
[1 0 1 1 0 0 0 0 0 0 1 1 0]
[0 0 1 0 0 0 0 1 1 1 1 0 0]
[1 1 0 1 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0 0]
[13, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 0 0 1 1 1 0 1 0]
[0 1 0 0 0 0 1 1 0 0 0 1 0]
[0 0 1 0 0 0 0 1 1 1 1 0 0]
[0 0 0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0 0]
hull dim 3
[ <0, 1>, <4, 6>, <5, 5>, <6, 8>, <7, 7>, <8, 1>, <9, 3>, <11, 1> ]
k=6
Construction I from a [11, 5, 4] code with h=2
6 %%%
[1 0 0 1 1 1 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 1 1 1 0 1]
[1 1 0 1 0 0 0 1 1 0 0 1 0]
[1 1 0 0 1 0 0 1 0 1 1 0 0]
[1 1 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
[13, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 0 1 1 1 1 0]
[0 1 0 0 0 0 0 1 0 1 0 1 0]
[0 0 1 0 0 0 0 0 1 1 1 0 1]
[0 0 0 1 0 1 0 0 0 0 1 1 0]
[0 0 0 0 1 1 0 0 1 1 0 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
hull dim 3
[ <0, 1>, <4, 10>, <5, 6>, <6, 16>, <7, 14>, <8, 5>, <9, 10>, <11, 2> ]
Construction IV from a [11, 5, 4] code with h=3
485 %%%
[1 0 0 0 1 1 0 0 0 0 0 1 1]
[1 1 1 0 0 0 0 1 1 0 0 1 0]
[0 0 0 1 0 0 0 0 1 1 1 0 0]
[1 1 0 0 1 0 0 1 0 1 1 0 0]
[1 1 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
[13, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 1 0 1 1]
[0 1 0 0 0 1 0 1 0 1 1 1 1]
[0 0 1 0 0 1 0 0 0 0 1 1 0]
[0 0 0 1 0 0 0 0 1 1 1 0 0]
[0 0 0 0 1 1 0 0 1 1 0 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
hull dim 3
[ <0, 1>, <4, 8>, <5, 8>, <6, 14>, <7, 16>, <8, 7>, <9, 8>, <10, 2> ]
k=7
Construction I from a [11, 6, 4] code with h=2
[13, 7, 4]
5 %%%
[1 0 1 0 1 1 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 1 0 1 0 1]
[0 0 0 1 0 0 0 0 1 1 0 0 1]
[1 1 0 0 1 0 0 0 0 1 1 1 0]
[1 1 0 0 0 1 0 0 1 0 1 1 0]
[0 0 0 0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 0 0 0 1 1 1 1 0 0]
[13, 7, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 0 1 1 0 1 1]
[0 1 0 0 0 0 0 0 0 1 1 0 1]
[0 0 1 0 0 1 0 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 1 1 0 0 1]
[0 0 0 0 1 1 0 0 1 1 0 0 0]
[0 0 0 0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 0 0 0 1 1 1 1 0 0]
hull dim 3
[ <0, 1>, <4, 27>, <6, 44>, <8, 51>, <10, 4>, <12, 1> ]
k=8
Construction I from a [11, 7, 3] code with h=2
[13, 8, 3]
33 %%%
[1 0 1 0 0 0 0 1 1 0 0 0 0]
[1 1 1 0 0 0 0 0 0 1 1 0 1]
[0 0 0 1 0 0 0 0 0 0 1 0 1]
[0 0 0 0 1 0 0 0 0 1 0 0 1]
[0 0 0 0 0 1 0 0 0 1 1 1 0]
[0 0 0 0 0 0 1 0 0 0 1 1 0]
[1 1 0 0 0 0 0 1 0 1 0 1 0]
[1 1 0 0 0 0 0 0 1 1 1 0 0]
[13, 8, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 0 1 1 1]
[0 1 0 0 0 0 0 0 0 1 0 1 1]
[0 0 1 0 0 0 0 0 1 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 1 0 1]
[0 0 0 0 1 0 0 0 0 1 0 0 1]
[0 0 0 0 0 1 0 0 0 1 1 1 0]
[0 0 0 0 0 0 1 0 0 0 1 1 0]
[0 0 0 0 0 0 0 1 1 0 1 1 0]
hull dim 3
[ <0, 1>, <3, 10>, <4, 24>, <5, 39>, <6, 54>, <7, 54>, <8, 39>, <9, 24>, <10,
10>, <13, 1> ]
k=9
Construction IV from a [11, 8, 2] code with h=3
1 %%%
[1 0 1 1 0 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 1 1 1]
[1 1 0 1 0 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 0 1 1 0 0]
[13, 9, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 1 0 0]
[0 1 0 1 0 0 0 0 0 0 1 1 1]
[0 0 1 1 0 0 0 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 0 1 1 0 0]
hull dim 3
[ <0, 1>, <2, 15>, <3, 18>, <4, 22>, <5, 84>, <6, 106>, <7, 104>, <8, 105>, <9,
44>, <10, 7>, <11, 6> ]
k=10
Construction I from a [11, 9, 1] code with h=2
1 %%%
[1 0 1 1 1 1 1 1 1 1 1 0 0]
[1 1 1 0 0 0 0 0 0 0 0 0 1]
[1 1 0 1 0 0 0 0 0 0 0 1 0]
[1 1 0 0 1 0 0 0 0 0 0 0 0]
[1 1 0 0 0 1 0 0 0 0 0 0 0]
[1 1 0 0 0 0 1 0 0 0 0 0 0]
[1 1 0 0 0 0 0 1 0 0 0 0 0]
[1 1 0 0 0 0 0 0 1 0 0 0 0]
[1 1 0 0 0 0 0 0 0 1 0 0 0]
[1 1 0 0 0 0 0 0 0 0 1 0 0]
[13, 10, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 1 1 1]
[0 1 0 0 0 0 0 0 0 0 0 1 1]
[0 0 1 0 0 0 0 0 0 0 1 0 1]
[0 0 0 1 0 0 0 0 0 0 1 1 0]
[0 0 0 0 1 0 0 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 0 1 1 0 0]
hull dim 3
[ <0, 1>, <2, 21>, <3, 25>, <4, 66>, <5, 189>, <6, 210>, <7, 210>, <8, 189>, <9,
66>, <10, 25>, <11, 21>, <13, 1> ]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
"h=4"
n= 12 (h=4)
k=4
Fact: There exist 10 self-orthogonal [12, 4, 4] codes by Bouyukliev, Bouyuklieva and Gulliver.
One of them is as follows:
C:=LinearCode
[1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 1, 1, 0, 1, 0 ,0, 0, 0],
[0, 0, 1, 0, 1, 1 ,0 ,0, 1 ,0 ,0 ,0],
[0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1]>;
> C;
[12, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 1 0 0 0 0 0]
[0 1 0 0 1 1 0 1 0 0 0 0]
[0 0 1 0 1 1 0 0 1 0 0 0]
[0 0 0 1 0 1 1 1 1 1 1 1]
> Hull(C);
[12, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 1 1 0 0 0 0 0]
[0 1 0 0 1 1 0 1 0 0 0 0]
[0 0 1 0 1 1 0 0 1 0 0 0]
[0 0 0 1 0 1 1 1 1 1 1 1]
> WeightDistribution(C);
[ <0, 1>, <4, 6>, <8, 9> ]
k=5
Construction I from a [10, 4, 4] code with h=3
249 %%%
[1 0 1 0 0 0 0 0 0 0 1 1]
[0 0 1 0 0 0 0 1 1 1 1 0]
[0 0 0 1 0 0 1 0 1 1 0 0]
[0 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0]
[12, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 1 1 0 1]
[0 0 1 0 0 0 0 1 1 1 1 0]
[0 0 0 1 0 0 1 0 1 1 0 0]
[0 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0]
hull dim 4
[ <0, 1>, <4, 8>, <5, 14>, <8, 7>, <9, 2> ]
k=6
Construction I from a [10,5,4] code with h=3
5 %%%
[1 0 1 0 1 1 0 0 0 0 0 0]
[1 1 1 0 0 0 0 1 1 0 0 1]
[0 0 0 1 0 0 0 0 1 1 1 0]
[1 1 0 0 1 0 0 1 0 1 1 0]
[1 1 0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 0 0 1 1 1 1 0 0]
[12, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 0 1 1 1 1]
[0 1 0 0 0 0 0 1 0 1 0 1]
[0 0 1 0 0 1 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 1 1 1 0]
[0 0 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0]
hull dim 4
[ <0, 1>, <4, 16>, <6, 30>, <8, 15>, <10, 2> ]
k=7
Construction I from a [10,6,2] code with h=3
1 %%%
[1 0 1 0 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 1 1 0 0 0]
[12, 7, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 1 1 0 0 0]
hull dim 4
[ <0, 1>, <2, 7>, <3, 4>, <4, 10>, <5, 28>, <6, 22>, <7, 28>, <8, 21>, <9, 4>,
<10, 3> ]
k=8
Construction I from a [10,7,2] code with h=3
1 %%%
[1 0 1 0 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0]
[12, 8, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0]
hull dim 4
[ <0, 1>, <2, 18>, <4, 63>, <6, 92>, <8, 63>, <10, 18>, <12, 1> ]
n= 13 (h=4)
k=4
Fact: There are exactly six inequivalent binary self-orthogonal [13, 4, 4] codes by
Bouyukliev, Bouyuklieva and Gulliver.
C:=LinearCode
[1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0],
[0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1]>;
> C;
[13, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 0 0 0 0 0]
[0 1 0 0 1 1 0 1 0 0 0 0 0]
[0 0 1 0 1 1 1 0 1 1 1 1 0]
[0 0 0 1 0 0 0 0 0 1 1 0 1]
> Hull(C);
[13, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 1 1 0 0 0 0 0]
[0 1 0 0 1 1 0 1 0 0 0 0 0]
[0 0 1 0 1 1 1 0 1 1 1 1 0]
[0 0 0 1 0 0 0 0 0 1 1 0 1]
> WeightDistribution(C);
[ <0, 1>, <4, 4>, <8, 11> ]
k=5
Construction I from a [11, 4, 4] code with h=3
249 %%%
[1 0 1 0 0 0 0 0 0 0 1 1 0]
[0 0 1 0 0 0 0 1 1 1 1 0 0]
[0 0 0 1 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0 0]
[13, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 1 1 0 1 0]
[0 0 1 0 0 0 0 1 1 1 1 0 0]
[0 0 0 1 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0 0]
hull dim 4
[ <0, 1>, <4, 8>, <5, 14>, <8, 7>, <9, 2> ]
Ccnstruction IV from a [11, 4, 4 ] code with h=4
242 %%%
[1 0 1 1 0 0 0 0 0 0 1 1 0]
[1 1 1 0 0 0 0 1 1 1 0 0 0]
[1 1 0 1 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0 0]
[13, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 1 1 0 0 1 1 0]
[0 1 0 1 0 0 0 1 1 1 1 1 0]
[0 0 1 1 0 0 1 1 0 0 0 0 0]
[0 0 0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0 0]
hull dim 4
[ <0, 1>, <4, 6>, <5, 4>, <6, 8>, <7, 8>, <8, 1>, <9, 4> ]
k=6
Construction I from a [11, 5, 4] code with h=3
5 %%% Construction I from a [11, 5, 4] code with h=3
[1 0 1 0 1 1 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 1 1 0 0 1 0]
[0 0 0 1 0 0 0 0 1 1 1 0 0]
[1 1 0 0 1 0 0 1 0 1 1 0 0]
[1 1 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
[13, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 0 1 1 1 1 0]
[0 1 0 0 0 0 0 1 0 1 0 1 0]
[0 0 1 0 0 1 0 0 0 0 1 1 0]
[0 0 0 1 0 0 0 0 1 1 1 0 0]
[0 0 0 0 1 1 0 0 1 1 0 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
hull dim 4
[ <0, 1>, <4, 16>, <6, 30>, <8, 15>, <10, 2> ]
Construction III from a [11, 5, 4] code with h=3
1024 %%%
[1 1 1 0 0 0 0 0 0 0 0 0 1]
[1 0 1 0 0 0 0 1 1 0 0 1 0]
[0 0 0 1 0 0 0 0 1 1 1 0 0]
[0 0 0 0 1 0 0 1 0 1 1 0 0]
[0 0 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
[13, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 1 1 0 0 1 0]
[0 1 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 1 0 0 0 0 1 1 1 0 0]
[0 0 0 0 1 0 0 1 0 1 1 0 0]
[0 0 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
hull dim 4
[ <0, 1>, <4, 15>, <5, 8>, <7, 16>, <8, 15>, <9, 8>, <12, 1> ]
Construction IV from a [11,5,4] code with h=4
5 %%%
[1 0 1 1 1 1 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 1 1 1 0 1]
[1 1 0 1 0 0 0 0 1 1 1 1 0]
[1 1 0 0 1 0 0 1 0 1 1 0 0]
[1 1 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
[13, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 1 0 1 1]
[0 1 0 0 0 1 0 1 0 1 1 1 1]
[0 0 1 0 0 1 0 1 0 1 0 0 1]
[0 0 0 1 0 1 0 1 0 1 0 1 0]
[0 0 0 0 1 1 0 0 1 1 0 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
hull dim 4
[ <0, 1>, <4, 6>, <5, 14>, <6, 12>, <7, 14>, <8, 9>, <9, 2>, <10, 4>, <11, 2> ]
k=7
Construction I from a [11, 6, 3] code with h=3
3 %%%
[1 0 1 1 1 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 1 0 0 0 1]
[1 1 0 1 0 0 0 0 1 0 0 1 0]
[1 1 0 0 1 0 0 0 1 1 1 0 0]
[0 0 0 0 0 1 0 0 0 1 1 0 0]
[0 0 0 0 0 0 1 0 1 0 1 0 0]
[0 0 0 0 0 0 0 1 1 1 0 0 0]
[13, 7, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 0 0 1 1 1 1 1]
[0 0 1 0 1 0 0 0 0 1 1 0 1]
[0 0 0 1 1 0 0 0 0 1 1 1 0]
[0 0 0 0 0 1 0 0 0 1 1 0 0]
[0 0 0 0 0 0 1 0 1 0 1 0 0]
[0 0 0 0 0 0 0 1 1 1 0 0 0]
hull dim 4
[ <0, 1>, <3, 4>, <4, 10>, <5, 21>, <6, 28>, <7, 28>, <8, 21>, <9, 10>, <10, 4>,
<13, 1> ]
Construction IV from a [11, 6, 3] code with h=4
4 %%%
[1 0 0 0 1 1 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 1 1 1 0 1]
[1 1 0 0 1 0 0 0 1 1 1 1 0]
[1 1 0 0 0 1 0 0 0 1 1 0 0]
[0 0 0 0 0 0 1 0 1 0 1 0 0]
[0 0 0 0 0 0 0 1 1 1 0 0 0]
[13, 7, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 0 0 1 0]
[0 1 0 0 0 1 0 0 1 1 1 1 0]
[0 0 1 0 0 0 0 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 1 1 1 0 1]
[0 0 0 0 1 1 0 0 1 0 0 1 0]
[0 0 0 0 0 0 1 0 1 0 1 0 0]
[0 0 0 0 0 0 0 1 1 1 0 0 0]
hull dim 4
[ <0, 1>, <3, 5>, <4, 10>, <5, 20>, <6, 28>, <7, 26>, <8, 21>, <9, 12>, <10, 4>,
<11, 1> ]
k=8
1 %%% Construction I from a [11, 7,2] code with h=3
[1 0 1 0 0 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 0 1 0]
[0 0 0 1 0 0 0 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0 0]
[13, 8, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 1 0 0 0 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0 0]
hull dim 4
[ <0, 1>, <2, 18>, <4, 63>, <6, 92>, <8, 63>, <10, 18>, <12, 1> ]
Construction IV from a [11,7,2] code with h=4
1 %%%
[1 0 1 1 0 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 1 1 1]
[1 1 0 1 0 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0 0]
[13, 8, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 1 0 0]
[0 1 0 1 0 0 0 0 0 0 1 1 1]
[0 0 1 1 0 0 0 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0 0]
hull dim 4
[ <0, 1>, <2, 7>, <3, 6>, <4, 14>, <5, 44>, <6, 50>, <7, 56>, <8, 49>, <9, 20>,
<10, 7>, <11, 2> ]
k=9
Construciton I from a [11, 8, 2] code with h=3
1 %%%
[1 0 1 0 0 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 1 1 1]
[0 0 0 1 0 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 0 1 1 0 0]
[13, 9, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 1 1 1]
[0 0 0 1 0 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 0 1 1 0 0]
hull dim 4
[ <0, 1>, <2, 11>, <3, 19>, <4, 38>, <5, 89>, <6, 98>, <7, 98>, <8, 89>, <9,
38>, <10, 19>, <11, 11>, <13, 1> ]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
"h=5"
n=12 (h=5)
k=5
Construction I from a [10,4,4] code with h=4
129 %%%
[1 0 1 0 0 0 0 0 0 1 1 0]
[0 0 1 0 0 0 0 1 1 1 0 0]
[1 1 0 1 0 0 1 0 1 1 0 0]
[1 1 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0]
[12, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 1 0 1 0]
[0 1 0 0 1 0 1 0 1 1 1 0]
[0 0 1 0 0 0 0 1 1 1 0 0]
[0 0 0 1 1 0 0 1 1 0 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0]
hull dim 5
[ <0, 1>, <4, 10>, <6, 16>, <8, 5> ]
k=6
Construction I from a [10,5,4] code with h=4
5 %%%
[1 0 1 0 1 1 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 1 1 1 1]
[0 0 0 1 0 0 0 1 0 1 1 1]
[1 1 0 0 1 0 0 1 1 0 0 1]
[1 1 0 0 0 1 0 1 1 0 1 0]
[0 0 0 0 0 0 1 1 1 1 0 0]
[12, 6, 3] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1 0 1 0 1 1 0]
[0 1 0 0 0 0 0 0 1 1 0 0]
[0 0 1 0 0 1 0 1 0 1 0 1]
[0 0 0 1 0 0 0 1 0 1 1 1]
[0 0 0 0 1 1 0 0 0 0 1 1]
[0 0 0 0 0 0 1 1 1 1 0 0]
hull dim 5
[ <0, 1>, <3, 2>, <4, 8>, <5, 14>, <6, 14>, <7, 14>, <8, 7>, <9, 2>, <10, 2> ]
k=7
Construction I from a [10,6,2] code with h=4
1 %%%
[1 0 1 0 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 1 1 0 0 0]
[12, 7, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 1 1 0 0 0]
hull dim 5
[ <0, 1>, <2, 10>, <4, 31>, <6, 44>, <8, 31>, <10, 10>, <12, 1> ]
n=13 (h=5)
k=5
Fact: There are 11 [13,5,4] SO codes but only five of them do not have zero coordinates.
by Bouyukliev, Bouyuklieva and Gulliver.
We construct one of them from building-up as follows.
Construction I from a [11,4,4] code with h=4
129 %%%
[1 0 1 0 0 0 0 0 0 1 1 0 0]
[0 0 1 0 0 0 0 1 1 1 0 0 0]
[1 1 0 1 0 0 1 0 1 1 0 0 0]
[1 1 0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0 0]
[13, 5, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 1 1 0 1 0 0]
[0 1 0 0 1 0 1 0 1 1 1 0 0]
[0 0 1 0 0 0 0 1 1 1 0 0 0]
[0 0 0 1 1 0 0 1 1 0 0 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0 0]
hull dim 5
[ <0, 1>, <4, 10>, <6, 16>, <8, 5> ]
k=6
Construction I from a [11, 5, 4] code with h=4
3 %%%
[1 0 1 1 1 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 1 1 1 0 1]
[1 1 0 1 0 0 0 0 1 1 1 1 0]
[1 1 0 0 1 0 0 1 0 1 1 0 0]
[0 0 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
[13, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0 0 0 0 0 0 1 1]
[0 1 0 0 0 0 0 1 0 1 1 1 1]
[0 0 1 0 1 0 0 1 1 0 0 0 1]
[0 0 0 1 1 0 0 1 1 0 0 1 0]
[0 0 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 1 1 1 1 0 0 0]
hull dim 5
[ <0, 1>, <4, 6>, <5, 12>, <6, 12>, <7, 16>, <8, 9>, <9, 4>, <10, 4> ]
k=7
Start from a self-dual [12, 6, 4] code B_{12} from Pless' paper, here denoted by G_12
G_12 :=LinearCode
[1,1,1,1,0,0,0,0,0,0,0,0],
[0,0,1,1,1,1,0,0,0,0,0,0],
[0,0,0,0,1,1,1,1,0,0,0,0],
[0,0,0,0,0,0,1,1,1,1,0,0],
[0,0,0,0,0,0,0,0,1,1,1,1],
[0,1,0,1,0,1,0,1,0,1,0,1]>;
> G_12;
[12, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 0 1 0 1 1 0]
[0 1 0 1 0 1 0 1 0 1 0 1]
[0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 1 1 0 0 0 0 1 1]
[0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1]
> Hull(G_12);
[12, 6, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 0 1 0 1 1 0]
[0 1 0 1 0 1 0 1 0 1 0 1]
[0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 1 1 0 0 0 0 1 1]
[0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1]
> WeightDistribution(G_12);
[ <0, 1>, <4, 15>, <6, 32>, <8, 15>, <12, 1> ]
>
%% Augment [1,0,0,0,0,0,0,0,1,0,1,0,1] to G_12 to get G_13
G_13 :=LinearCode
[1,0,0,0,0,0,0,0,1,0,1,0,1],
[0,1,1,1,1,0,0,0,0,0,0,0,0],
[0,0,0,1,1,1,1,0,0,0,0,0,0],
[0,0,0,0,0,1,1,1,1,0,0,0,0],
[0,0,0,0,0,0,0,1,1,1,1,0,0],
[0,0,0,0,0,0,0,0,0,1,1,1,1],
[0,0,1,0,1,0,1,0,1,0,1,0,1]>;
> G_13;
[13, 7, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 0 1 0 1]
[0 1 0 0 1 0 1 0 1 0 1 1 0]
[0 0 1 0 1 0 1 0 1 0 1 0 1]
[0 0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 0 1 1 0 0 0 0 1 1]
[0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1 1 1 1]
> Hull(G_13);
[13, 5, 4] Linear Code over GF(2)
Generator matrix:
[0 1 0 0 1 0 1 0 1 0 1 1 0]
[0 0 1 0 1 1 0 0 1 0 1 1 0]
[0 0 0 1 1 1 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1 1 1 1]
> WeightDistribution(G_13);
[ <0, 1>, <4, 23>, <6, 56>, <8, 39>, <10, 8>, <12, 1> ]
k=8
Construction I from a [11, 7, 2] code with h=4
1 %%%
[1 0 1 0 0 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 1 1 1]
[0 0 0 1 0 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0 0]
[13, 8, 2] Linear Code over GF(2)
Generator matrix:
[1 0 1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 1 1 1]
[0 0 0 1 0 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 1 0 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0 0]
hull dim 5
[ <0, 1>, <2, 7>, <3, 7>, <4, 14>, <5, 49>, <6, 50>, <7, 50>, <8, 49>, <9, 14>,
<10, 7>, <11, 7>, <13, 1> ]
// Construction III may give same rank as the bottom rank l, so that
// building up can derive hull dimension l, l+1, l+2. For example,
1
[1 1 0 0 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 1 0 1 1 0 0 0 0]
[0 0 0 1 0 1 1 0 1 0 0 0 0]
[0 0 0 0 1 1 1 1 0 0 0 0 0]
[13, 4, 2] Linear Code over GF(2)
Generator matrix:
[1 1 0 0 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 1 0 1 1 0 0 0 0]
[0 0 0 1 0 1 1 0 1 0 0 0 0]
[0 0 0 0 1 1 1 1 0 0 0 0 0]
hull dim 4
[ <0, 1>, <2, 1>, <4, 7>, <6, 7> ]
2
[1 1 1 1 0 0 0 0 0 0 0 0 0]
[1 0 1 0 0 1 0 1 1 0 0 0 0]
[1 0 0 1 0 1 1 0 1 0 0 0 0]
[0 0 0 0 1 1 1 1 0 0 0 0 0]
[13, 4, 4] Linear Code over GF(2)
Generator matrix:
[1 0 0 1 0 1 1 0 1 0 0 0 0]
[0 1 0 1 0 1 0 1 1 0 0 0 0]
[0 0 1 1 0 0 1 1 0 0 0 0 0]
[0 0 0 0 1 1 1 1 0 0 0 0 0]
hull dim 3
[ <0, 1>, <4, 6>, <5, 8>, <8, 1> ]