The generator matrix
1 0 1 X
0 1 1 X^2
0 0 X X
0 0 0 X^3
generates a code of length 4 over Z2[X]/(X^4) who´s minimum homogenous weight is 2.
Homogenous weight enumerator: w(x)=1x^0+124x^2+768x^3+2310x^4+768x^5+124x^6+1x^8
The gray image is a linear code over GF(2) with n=32, k=12 and d=8.
As d=10 is an upper bound for linear (32,12,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 12.
This code was found by Heurico 1.16 in -1.01e-007 seconds.