The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  X  X  1  1  X  X  1  1  X  X  1  1  1  0  1  X  X  X  X  X  1
 0  X  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  X  X  X  X  0  0  X  X  X  0  X  X  X  0  X  0  0  X  X  X  0  0  X  X  0  X  X  X  0  0  X  0  X  0  X  X  X  X
 0  0  X  0  0  0  0  0  0  0  0  0  X  0  0  X  X  X  X  X  X  0  0  0  0  0  X  X  X  X  X  0  X  X  0  0  0  X  X  0  X  X  0  X  0  X  X  X  0  0  X  X  X  X  X
 0  0  0  X  0  0  0  0  0  0  0  0  X  X  X  X  X  0  X  X  X  0  X  X  X  0  X  0  0  X  X  X  0  0  0  X  0  X  X  0  0  0  0  0  X  X  X  X  X  0  0  0  X  0  X
 0  0  0  0  X  0  0  0  0  0  0  X  0  X  X  0  X  X  X  0  X  X  X  0  0  0  0  0  X  0  0  0  X  0  0  X  X  X  0  X  X  0  X  X  0  0  X  X  0  X  X  0  0  0  X
 0  0  0  0  0  X  0  0  0  0  0  X  0  0  X  X  X  X  X  0  0  X  0  0  X  X  0  0  0  X  X  0  0  0  X  0  X  0  X  X  0  X  X  0  0  0  X  X  0  0  X  X  X  0  X
 0  0  0  0  0  0  X  0  0  0  X  0  0  0  X  X  0  0  X  0  0  X  X  0  X  0  X  0  X  0  X  0  X  X  X  0  0  X  0  0  0  0  X  X  0  X  0  X  X  X  0  0  0  0  X
 0  0  0  0  0  0  0  X  0  0  X  0  0  X  0  0  X  0  0  X  0  X  X  0  0  X  X  X  0  0  X  X  X  X  X  0  0  0  0  X  X  X  0  X  0  0  0  0  0  X  0  X  0  X  0
 0  0  0  0  0  0  0  0  X  0  X  X  X  X  0  0  0  0  X  0  0  0  X  X  X  X  0  0  0  X  0  X  0  X  0  X  0  0  0  X  X  0  X  0  0  0  X  0  X  X  X  X  0  X  X
 0  0  0  0  0  0  0  0  0  X  X  X  X  0  0  X  X  X  X  X  0  X  0  X  0  X  0  X  X  0  0  X  X  0  X  0  X  X  0  0  X  X  0  0  0  0  0  0  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+112x^44+8x^46+246x^48+90x^50+321x^52+224x^54+388x^56+148x^58+256x^60+40x^62+133x^64+2x^66+62x^68+16x^72+1x^84

The gray image is a linear code over GF(2) with n=110, k=11 and d=44.
This code was found by Heurico 1.16 in 0.776 seconds.