The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 39.

Homogenous weight enumerator: w(x)=1x^0+58x^39+172x^40+366x^41+439x^42+682x^43+688x^44+662x^45+492x^46+334x^47+95x^48+58x^49+25x^50+14x^51+4x^52+2x^53+1x^54+3x^62

The gray image is a linear code over GF(2) with n=352, k=12 and d=156.
This code was found by Heurico 1.16 in 0.188 seconds.