The generator matrix

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

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+335x^40+96x^41+808x^42+480x^43+870x^44+288x^45+704x^46+160x^47+311x^48+24x^50+8x^52+9x^56+2x^60

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 3.7 seconds.