The generator matrix

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

generates a code of length 22 over Z2[X]/(X^2) who´s minimum homogenous weight is 13.

Homogenous weight enumerator: w(x)=1x^0+26x^13+129x^14+244x^15+484x^16+646x^17+944x^18+1236x^19+1513x^20+1912x^21+1958x^22+1948x^23+1680x^24+1312x^25+932x^26+636x^27+398x^28+190x^29+129x^30+32x^31+19x^32+10x^33+4x^34+1x^36

The gray image is a linear code over GF(2) with n=44, k=14 and d=13.
This code was found by Heurico 1.16 in 8.81 seconds.