The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 X X X X X X X 1 1 X^2 X^2 X^2 X^2 X^2 X^2 X^2 X X^2
0 X^3 0 0 0 X^3 X^3 X^3 0 0 0 X^3 0 X^3 0 X^3 X^3 0 X^3 X^3 0 X^3 X^3 0 0 X^3 X^3 0 X^3 X^3 0 0
0 0 X^3 0 X^3 X^3 X^3 0 0 0 X^3 X^3 X^3 X^3 X^3 X^3 0 X^3 X^3 0 0 0 0 0 X^3 X^3 0 X^3 X^3 0 0 0
0 0 0 X^3 X^3 0 X^3 X^3 0 X^3 X^3 0 0 X^3 X^3 0 0 0 X^3 X^3 X^3 X^3 0 X^3 X^3 0 0 0 X^3 X^3 0 0
generates a code of length 32 over Z2[X]/(X^4) who´s minimum homogenous weight is 32.
Homogenous weight enumerator: w(x)=1x^0+123x^32+4x^40
The gray image is a linear code over GF(2) with n=256, k=7 and d=128.
As d=128 is an upper bound for linear (256,7,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 7.
This code was found by Heurico 1.16 in 0.016 seconds.