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

generates a code of length 44 over Z2[X]/(X^2) who´s minimum homogenous weight is 35.

Homogenous weight enumerator: w(x)=1x^0+56x^35+107x^36+160x^37+217x^38+250x^39+283x^40+284x^41+271x^42+282x^43+315x^44+290x^45+307x^46+270x^47+240x^48+234x^49+178x^50+150x^51+65x^52+54x^53+43x^54+16x^55+11x^56+2x^57+7x^58+1x^60+1x^62+1x^64

The gray image is a linear code over GF(2) with n=88, k=12 and d=35.
This code was found by Heurico 1.16 in 1.53 seconds.