The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  0  X  1  1  1  1  1  1  1  X  1  1  1  1  1 X^3+X^2  1  1 X^2  1  1  1  1  X  1  X  1  1  1 X^3  1  1  1
 0  X  0 X^3+X^2+X X^2 X^2+X X^3+X^2  X X^3  0 X^2+X X^2+X X^2 X^3+X X^2  X X^2+X  X X^3+X  0 X^3+X^2+X  0 X^2+X X^2+X  X X^3+X^2 X^3+X^2+X X^2+X X^3  0  X X^3+X  X X^3+X^2 X^3+X^2  X X^3+X^2 X^3+X^2 X^3+X^2  0 X^3+X^2+X X^2 X^3  X X^3+X^2+X X^3  0 X^2+X X^2+X X^3
 0  0 X^3+X^2  0 X^2  0  0 X^3  0 X^2 X^2 X^2 X^2 X^3+X^2 X^3 X^2 X^3+X^2 X^3 X^2 X^3  0  0 X^3 X^2 X^3 X^3  0 X^3+X^2 X^3+X^2  0 X^3+X^2 X^3+X^2  0 X^2 X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^2  0 X^3+X^2 X^3 X^3 X^3 X^2  0 X^2  0
 0  0  0 X^3+X^2  0  0 X^3 X^2 X^2 X^2 X^2 X^3 X^3+X^2  0 X^2 X^2 X^3+X^2  0 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3  0  0 X^3 X^2 X^2 X^3+X^2 X^3+X^2 X^3 X^3+X^2 X^3 X^3  0 X^3 X^2 X^3+X^2 X^3 X^3+X^2 X^3 X^3+X^2 X^2 X^3+X^2 X^3 X^3+X^2 X^2 X^3 X^2 X^3+X^2
 0  0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3  0 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  0  0 X^3 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3  0

generates a code of length 50 over Z2[X]/(X^4) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+52x^44+102x^45+162x^46+258x^47+614x^48+418x^49+956x^50+470x^51+568x^52+172x^53+136x^54+74x^55+35x^56+26x^57+21x^58+10x^59+10x^60+2x^61+4x^62+4x^63+1x^82

The gray image is a linear code over GF(2) with n=400, k=12 and d=176.
This code was found by Heurico 1.16 in 0.375 seconds.