The generator matrix

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

generates a code of length 53 over Z2[X]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+184x^44+348x^46+503x^48+476x^50+581x^52+546x^54+526x^56+394x^58+288x^60+138x^62+80x^64+18x^66+11x^68+2x^72

The gray image is a linear code over GF(2) with n=106, k=12 and d=44.
This code was found by Heurico 1.16 in 1.9 seconds.