The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 X^2  1  1  1  1  1  1  0  0 X^3+X^2  1  1
 0  X X^3+X^2 X^2+X  0 X^2+X X^3+X^2 X^3+X X^2+X  0 X^3+X X^3+X^2 X^3 X^3+X^2+X X^2 X^3+X  0 X^2+X X^3+X^2 X^3+X  0 X^2+X  0 X^3+X^2+X X^3+X^2 X^2 X^3+X X^3+X X^3 X^2+X X^2  X X^3+X^2  0 X^3 X^2+X X^3+X^2+X X^3+X^2 X^3+X^2 X^2  X  X X^3+X^2 X^2
 0  0 X^3  0  0  0 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3  0  0  0 X^3
 0  0  0 X^3  0  0  0  0  0  0  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0  0  0  0  0  0 X^3 X^3
 0  0  0  0 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0  0  0 X^3 X^3  0 X^3 X^3  0
 0  0  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3  0  0 X^3 X^3  0  0  0  0  0 X^3 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+204x^40+296x^42+256x^43+574x^44+256x^45+240x^46+202x^48+8x^50+10x^52+1x^80

The gray image is a linear code over GF(2) with n=352, k=11 and d=160.
This code was found by Heurico 1.16 in 0.125 seconds.