The generator matrix

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

generates a code of length 22 over Z2[X]/(X^2) who´s minimum homogenous weight is 12.

Homogenous weight enumerator: w(x)=1x^0+96x^12+40x^14+234x^16+136x^18+640x^19+352x^20+336x^22+1280x^23+264x^24+336x^26+128x^27+64x^28+136x^30+13x^32+40x^34

The gray image is a linear code over GF(2) with n=44, k=12 and d=12.
This code was found by Heurico 1.16 in 0.337 seconds.