The generator matrix

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

generates a code of length 44 over Z2[X]/(X^2) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+45x^34+52x^35+111x^36+138x^37+170x^38+198x^39+276x^40+284x^41+284x^42+348x^43+316x^44+352x^45+287x^46+304x^47+219x^48+196x^49+177x^50+112x^51+83x^52+54x^53+52x^54+10x^55+14x^56+6x^58+2x^60+3x^62+2x^64

The gray image is a linear code over GF(2) with n=88, k=12 and d=34.
This code was found by Heurico 1.16 in 1.46 seconds.