The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 X^2  1  X  X  1  1  1 X^2  1  1  1  1  0  1  0  1  0  X  1 X^3+X^2  1  1  1  1  X
 0  X  0  X  0 X^3 X^3+X  X X^2 X^2+X X^2 X^3+X^2+X X^2 X^3+X^2 X^2+X X^2+X X^2 X^3+X^2+X X^3  X X^3  X  0 X^3  X X^2+X  X  X X^3+X X^2  X X^3 X^2  X X^3  X  X X^2+X  X X^2 X^3 X^3 X^3+X X^2+X
 0  0  X  X X^3+X^2 X^3+X^2+X X^2+X X^2 X^2 X^3+X^2+X X^3 X^3+X  X X^2+X  0 X^3+X^2  X  0 X^2+X X^3+X^2 X^3+X^2 X^3+X X^2+X X^3 X^2  X  X X^3+X X^3+X^2+X  0 X^3+X^2+X  X  X X^3+X^2+X X^3+X  0 X^3 X^2 X^2+X X^2+X X^2 X^3+X  0 X^3+X^2
 0  0  0 X^3  0  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3  0  0  0  0  0 X^3 X^3 X^3 X^3
 0  0  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0  0  0  0  0 X^3  0  0 X^3  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0 X^3  0  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+387x^40+104x^41+732x^42+448x^43+1020x^44+336x^45+540x^46+128x^47+265x^48+8x^49+68x^50+51x^52+4x^54+3x^56+1x^68

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 150 seconds.