The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 41.

Homogenous weight enumerator: w(x)=1x^0+466x^41+794x^42+678x^43+685x^44+482x^45+350x^46+270x^47+193x^48+140x^49+22x^50+12x^51+1x^52+2x^54

The gray image is a linear code over GF(2) with n=352, k=12 and d=164.
This code was found by Heurico 1.16 in 0.141 seconds.