The generator matrix

1 0 0 0 0 0 1 1 1 1 1 0
0 1 0 0 0 0 0 1 0 X 1 1
0 0 1 0 0 0 0 1 1 X+1 X 0
0 0 0 1 0 0 0 1 X 1 X+1 0
0 0 0 0 1 0 1 0 X+1 X X 1
0 0 0 0 0 1 1 X+1 X 0 0 1
0 0 0 0 0 0 X X 0 0 X 0

generates a code of length 12 over Z2[X]/(X^2) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+164x^6+703x^8+1820x^10+2816x^12+1820x^14+703x^16+164x^18+1x^24

The gray image is a linear code over GF(2) with n=24, k=13 and d=6.
As d=6 is an upper bound for linear (24,13,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 13.
This code was found by Heurico 1.10 in 0.078 seconds.