The generator matrix
1 0 1 1 1 X 1 1 0 1 1 X 1 1 0 1 1 X 1 1 1 1 0 X 1 1 1 1 0 X 1 1 1 1 0 X 1 1 1 1 X X 0 0 X X X 0 1 1 1 1 0 X X X 0 1 1 1 1 X X 0 0 X 1 1
0 1 X+1 X 1 1 0 X+1 1 X 1 1 0 X+1 1 X 1 1 0 X X+1 1 1 1 0 X X+1 1 1 1 0 X X+1 1 1 1 0 X X+1 1 0 X X 1 1 0 X X 0 X X+1 1 1 1 0 X X 0 X X+1 1 0 X X 1 1 0 X
generates a code of length 68 over Z2[X]/(X^2) who´s minimum homogenous weight is 72.
Homogenous weight enumerator: w(x)=1x^0+13x^72+2x^76
The gray image is a linear code over GF(2) with n=136, k=4 and d=72.
As d=72 is an upper bound for linear (136,4,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.0681 seconds.