The generator matrix

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

generates a code of length 55 over Z2[X]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+70x^44+114x^45+169x^46+296x^47+380x^48+340x^49+411x^50+494x^51+484x^52+512x^53+518x^54+564x^55+542x^56+574x^57+539x^58+462x^59+428x^60+384x^61+312x^62+202x^63+125x^64+118x^65+85x^66+28x^67+18x^68+6x^69+8x^70+2x^71+4x^74+1x^78+1x^90

The gray image is a linear code over GF(2) with n=110, k=13 and d=44.
This code was found by Heurico 1.16 in 7.71 seconds.