The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+157x^40+378x^41+684x^42+564x^43+746x^44+528x^45+502x^46+248x^47+152x^48+50x^49+42x^50+20x^51+16x^52+4x^53+3x^54+1x^62

The gray image is a linear code over GF(2) with n=352, k=12 and d=160.
This code was found by Heurico 1.16 in 0.187 seconds.