The generator matrix

 1  0  1  1  1 X^3+X^2+X  1  1  X  1 X^3+X^2  1  1  1  1 X^3+X^2+X  1  0  1  1 X^2+X  1 X^2  1 X^3+X  1 X^2  1  0 X^3+X^2+X  1  1 X^3+X^2  1  1  1 X^3  1  1  1  1  1 X^3+X^2  0
 0  1 X+1 X^3+X^2+X X^2+1  1 X^3+X^2+1 X^3+X^2  1  X  1 X^3+1 X^2+X+1 X+1  0  1 X^3+X^2+X  1 X+1 X^3  1 X^3+X  1 X^2  1 X^3+X^2+X  1 X^3+X+1  1  1 X^3+X^2+1 X^3+X^2+1  1 X^2  0  1  1 X^2+X+1 X^2+1 X^3+X^2+X+1  0 X^3  X  0
 0  0 X^2  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^2 X^2 X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3 X^3  0 X^3 X^3  0 X^3 X^2 X^3+X^2 X^3 X^2 X^3 X^3+X^2 X^3+X^2 X^3 X^3+X^2 X^3+X^2  0 X^3  0 X^2
 0  0  0 X^3+X^2 X^3 X^3+X^2 X^2 X^3 X^2 X^2 X^3  0 X^3 X^2 X^3+X^2 X^3  0 X^2 X^3+X^2 X^3 X^2 X^2 X^3 X^3+X^2 X^3  0 X^3+X^2 X^3+X^2 X^2  0  0 X^2 X^2 X^2 X^2  0  0 X^3+X^2 X^3+X^2  0 X^3+X^2 X^3+X^2 X^3+X^2 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+329x^40+216x^41+568x^42+640x^43+783x^44+384x^45+608x^46+288x^47+218x^48+8x^49+32x^50+5x^52+8x^54+8x^56

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 0.172 seconds.