The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X X X X X X X 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 X X X X 0 0 0 X 0 X X X 0 0 0 X 1 1 1 1 X
0 X 0 0 0 X X X 0 0 0 X 0 X X X 0 0 0 X 0 X X X 0 0 X X 0 X X X X X X 0 0 0 0 0 0 X 0 X X X 0 0 X 0 X 0 0 X X 0 X X X X 0 0 0 0 0 X X
0 0 X 0 X X X 0 0 0 X X X X 0 0 0 0 X X X X 0 0 0 X X 0 X X 0 0 0 X X X X 0 0 0 X X X X 0 0 0 X X X X X X X X 0 0 0 0 0 0 0 0 0 X X 0
0 0 0 X X 0 X X 0 X X 0 0 X X 0 0 X X 0 0 X X 0 X X 0 0 0 X X 0 X X 0 0 X X 0 X X 0 0 X X 0 X X 0 0 0 0 X X X 0 X 0 0 X X 0 0 X X 0 0
generates a code of length 67 over Z2[X]/(X^2) who´s minimum homogenous weight is 68.
Homogenous weight enumerator: w(x)=1x^0+18x^68+8x^70+5x^72
The gray image is a linear code over GF(2) with n=134, k=5 and d=68.
As d=68 is an upper bound for linear (134,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.0675 seconds.