The generator matrix

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

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+64x^44+88x^45+180x^46+298x^47+354x^48+434x^49+410x^50+464x^51+470x^52+492x^53+553x^54+534x^55+624x^56+528x^57+492x^58+520x^59+392x^60+396x^61+268x^62+190x^63+177x^64+94x^65+73x^66+40x^67+26x^68+16x^69+7x^70+2x^71+4x^72+1x^82

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 7.69 seconds.