The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  X  1  X  1  X  1  X X^2 X^2
 0 X^3+X^2  0  0  0 X^2 X^3+X^2 X^2  0 X^3 X^3  0 X^2 X^2 X^3+X^2 X^3+X^2  0 X^3  0  0 X^2 X^2 X^2 X^2 X^3 X^2 X^3 X^2  0 X^3 X^3+X^2 X^3 X^2 X^2 X^3  0 X^3 X^3+X^2  0 X^2  0 X^2 X^2 X^3+X^2
 0  0 X^3+X^2  0 X^2 X^2 X^2 X^3 X^3+X^2 X^3  0 X^2 X^2 X^2  0  0  0 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0 X^3 X^3+X^2 X^3+X^2 X^3 X^2  0 X^2 X^3+X^2 X^2 X^2 X^2 X^3+X^2 X^3 X^2  0 X^3+X^2  0 X^3+X^2 X^3 X^2 X^2 X^2
 0  0  0 X^3+X^2 X^2 X^3 X^3+X^2 X^3+X^2  0  0 X^2 X^2  0 X^3+X^2 X^2 X^3  0 X^3+X^2 X^2 X^3 X^2  0 X^2 X^3  0  0 X^3 X^3  0 X^2 X^2 X^3+X^2 X^3+X^2 X^3 X^3 X^3  0 X^3+X^2 X^3+X^2  0 X^3+X^2  0 X^3 X^2
 0  0  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0  0  0 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3 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+223x^40+192x^42+256x^43+764x^44+256x^45+192x^46+109x^48+44x^52+10x^56+1x^72

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 12.8 seconds.