The generator matrix

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

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+134x^44+1006x^45+2863x^46+6692x^47+11291x^48+20320x^49+29666x^50+38010x^51+41182x^52+39284x^53+29664x^54+20744x^55+11412x^56+5904x^57+2408x^58+1034x^59+356x^60+110x^61+31x^62+10x^63+8x^64+6x^66+4x^67+2x^70+2x^71

The gray image is a linear code over GF(2) with n=416, k=18 and d=176.
This code was found by Heurico 1.16 in 404 seconds.