The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  1  1 X^2 X^2
 0  X  0  0  0  0  0  0 X^2 X^2  X X^2+X X^2+X  X X^2+X X^2+X X^2  0 X^2+X  X  X X^2 X^2+X  0 X^2  0 X^2  X X^2
 0  0  X  0  0  0  0  0  0  0  0  0  0 X^2  X  X X^2+X X^2+X  X  X X^2 X^2+X X^2+X X^2 X^2+X X^2 X^2  X  X
 0  0  0  X  0  0 X^2 X^2+X  X  X  X  X X^2+X X^2  0  X  X X^2 X^2+X X^2+X  X  X  0  0 X^2  X  X X^2+X X^2
 0  0  0  0  X  0 X^2+X X^2+X  X X^2 X^2+X X^2+X X^2+X  0 X^2  X X^2+X  X X^2+X  0  0 X^2 X^2  X  0  0  X X^2 X^2+X
 0  0  0  0  0  X  X X^2 X^2+X  X X^2+X X^2  0  X  X  X  0  X X^2  X X^2+X  0  X X^2 X^2+X  X  X  X  0

generates a code of length 29 over Z2[X]/(X^3) who´s minimum homogenous weight is 22.

Homogenous weight enumerator: w(x)=1x^0+216x^22+495x^24+634x^26+384x^27+1696x^28+1280x^29+1746x^30+384x^31+723x^32+430x^34+151x^36+46x^38+5x^40+1x^52

The gray image is a linear code over GF(2) with n=116, k=13 and d=44.
This code was found by Heurico 1.16 in 3.24 seconds.