The generator matrix

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

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+60x^13+129x^14+222x^15+433x^16+650x^17+916x^18+1242x^19+1562x^20+1938x^21+2042x^22+1854x^23+1678x^24+1334x^25+912x^26+606x^27+396x^28+234x^29+93x^30+44x^31+24x^32+8x^33+4x^34+2x^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 10.3 seconds.