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  X  1  1  1  1  1  1  1  X  1  1  1  1  1  1  X  X  X  1  1  X  X  X  1  1
 0 X^3  0  0  0  0  0  0  0  0 X^3 X^3  0  0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3  0  0  0 X^3  0
 0  0 X^3  0  0  0  0  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3  0  0  0  0 X^3  0  0 X^3  0
 0  0  0 X^3  0  0  0  0  0 X^3  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0  0  0  0 X^3  0 X^3 X^3  0 X^3 X^3  0  0 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0
 0  0  0  0 X^3  0  0  0 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3  0 X^3  0  0  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0  0 X^3 X^3 X^3 X^3  0  0
 0  0  0  0  0 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0  0 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3  0  0 X^3 X^3  0 X^3 X^3  0 X^3  0  0 X^3
 0  0  0  0  0  0 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0  0 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0  0 X^3  0  0  0 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3  0
 0  0  0  0  0  0  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3  0  0  0 X^3 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0  0  0

generates a code of length 52 over Z2[X]/(X^4) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+41x^44+10x^46+107x^48+50x^50+1652x^52+62x^54+71x^56+6x^58+23x^60+20x^64+4x^68+1x^88

The gray image is a linear code over GF(2) with n=416, k=11 and d=176.
This code was found by Heurico 1.16 in 0.172 seconds.