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  X  1  X  1  X  1  1  1  1  1  X  1  1  1  1  X  1  X  X  X  1  1  X  1  1  X
 0  X  0  0  0  0  0  0  0  0  0  X  0  0  X  X  0  0  0  0  X  X  X  X  0  X  X  0  X  X  X  0  X  0  X  0  0  X  X  0  X  X  0  X  0  0  X  X  0  X  X  0
 0  0  X  0  0  0  0  0  0  0  0  X  X  X  X  X  0  0  0  X  X  0  0  0  0  0  0  0  0  0  0  X  X  X  X  X  X  0  X  X  0  X  0  X  0  X  0  X  0  X  0  X
 0  0  0  X  0  0  0  0  0  0  X  0  X  X  0  X  X  0  X  0  X  X  X  X  0  X  0  X  0  0  0  X  0  X  X  0  X  X  X  X  X  0  0  X  0  0  X  0  X  0  0  0
 0  0  0  0  X  0  0  0  0  0  X  0  0  X  X  X  X  X  0  X  0  0  0  0  0  X  X  X  0  0  X  0  X  X  X  0  0  0  X  0  X  0  0  0  X  X  X  0  0  X  0  X
 0  0  0  0  0  X  0  0  0  X  0  0  0  X  X  0  X  0  X  X  X  0  0  X  0  X  X  0  0  X  X  0  0  0  X  X  X  0  X  X  X  X  X  0  X  0  0  X  X  0  0  0
 0  0  0  0  0  0  X  0  0  X  0  0  X  0  0  X  X  X  0  X  X  0  X  X  0  0  X  0  X  0  0  0  0  X  0  X  0  X  X  X  0  0  0  0  X  0  X  X  0  X  0  X
 0  0  0  0  0  0  0  X  0  X  X  X  X  0  0  0  0  0  X  X  X  X  X  0  X  X  X  0  0  0  0  0  0  0  X  0  0  X  0  X  X  X  X  X  X  X  X  0  0  0  X  0
 0  0  0  0  0  0  0  0  X  X  X  X  0  0  X  X  X  0  0  0  X  0  0  0  X  0  0  0  X  X  0  X  X  0  0  X  0  X  0  X  X  X  0  X  X  X  X  X  X  0  X  X

generates a code of length 52 over Z2[X]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+124x^44+18x^46+178x^48+120x^50+207x^52+92x^54+140x^56+24x^58+72x^60+2x^62+33x^64+12x^68+1x^84

The gray image is a linear code over GF(2) with n=104, k=10 and d=44.
This code was found by Heurico 1.16 in 93.5 seconds.