The generator matrix

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

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+28x^44+98x^45+153x^46+166x^47+212x^48+230x^49+276x^50+266x^51+268x^52+262x^53+236x^54+300x^55+260x^56+278x^57+229x^58+194x^59+181x^60+128x^61+105x^62+84x^63+63x^64+28x^65+22x^66+12x^67+10x^68+2x^70+2x^71+1x^74+1x^76

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 1.82 seconds.