The generator matrix

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

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+188x^6+578x^8+1268x^10+1304x^12+580x^14+165x^16+12x^18

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 1.16 in 4.46 seconds.