The generator matrix

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

generates a code of length 22 over Z2[X]/(X^2) who´s minimum homogenous weight is 14.

Homogenous weight enumerator: w(x)=1x^0+79x^14+114x^15+237x^16+340x^17+461x^18+628x^19+770x^20+932x^21+988x^22+984x^23+800x^24+700x^25+456x^26+300x^27+222x^28+76x^29+61x^30+22x^31+18x^32+3x^34

The gray image is a linear code over GF(2) with n=44, k=13 and d=14.
This code was found by Heurico 1.16 in 1.5 seconds.