The generator matrix

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

generates a code of length 54 over Z2[X]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+103x^44+252x^46+408x^48+466x^50+530x^52+600x^54+552x^56+516x^58+315x^60+188x^62+100x^64+26x^66+28x^68+10x^72+1x^80

The gray image is a linear code over GF(2) with n=108, k=12 and d=44.
This code was found by Heurico 1.16 in 2.06 seconds.