The generator matrix

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

generates a code of length 22 over Z2[X]/(X^2) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+343x^16+404x^18+848x^20+844x^22+927x^24+492x^26+176x^28+52x^30+8x^32+1x^40

The gray image is a linear code over GF(2) with n=44, k=12 and d=16.
As d=16 is an upper bound for linear (44,12,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 12.
This code was found by Heurico 1.16 in 99.3 seconds.