The generator matrix

 1  0  0  0  1  1  1  1  1  1  1
 0  1  0  0  1  0  0  1  1  0  0
 0  0  1  0  1  0  1  0  1  0  0
 0  0  0  1  1  1  0  0  1  0  0
 0  0  0  0  X  0  0  0  0  0  X
 0  0  0  0  0  X  0  0  0  0  X
 0  0  0  0  0  0  X  0  0  0  X
 0  0  0  0  0  0  0  X  0  0  X
 0  0  0  0  0  0  0  0  X  0  X
 0  0  0  0  0  0  0  0  0  X  X

generates a code of length 11 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+103x^4+464x^6+2618x^8+4880x^10+5342x^12+2288x^14+629x^16+48x^18+11x^20

The gray image is a linear code over GF(2) with n=22, k=14 and d=4.
As d=4 is an upper bound for linear (22,14,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 14.
This code was found by Heurico 1.16 in 0.804 seconds.