The generator matrix

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

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+366x^44+1780x^45+3407x^46+5292x^47+7627x^48+9034x^49+10479x^50+9400x^51+7708x^52+5276x^53+2955x^54+1356x^55+513x^56+210x^57+77x^58+32x^59+16x^60+4x^61+2x^62+1x^64

The gray image is a linear code over GF(2) with n=400, k=16 and d=176.
This code was found by Heurico 1.16 in 34.6 seconds.