The generator matrix

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

generates a code of length 11 over Z2[X]/(X^2) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+184x^6+595x^8+1242x^10+1319x^12+580x^14+164x^16+10x^18+1x^20

The gray image is a linear code over GF(2) with n=22, k=12 and d=6.
As d=6 is an upper bound for linear (22,12,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 12.
This code was found by Heurico 1.10 in 0.172 seconds.