The generator matrix

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

generates a code of length 31 over Z2[X]/(X^3) who´s minimum homogenous weight is 23.

Homogenous weight enumerator: w(x)=1x^0+56x^23+175x^24+194x^25+242x^26+430x^27+649x^28+818x^29+955x^30+1062x^31+1045x^32+862x^33+613x^34+458x^35+278x^36+166x^37+98x^38+42x^39+27x^40+8x^41+8x^42+1x^44+3x^46+1x^50

The gray image is a linear code over GF(2) with n=124, k=13 and d=46.
This code was found by Heurico 1.16 in 8.99 seconds.