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  1  1  X  X  X
 0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2 X^3+X^2  0 X^2 X^3 X^3+X^2  0 X^3 X^2  0 X^2 X^2 X^3  0 X^3 X^3+X^2 X^3+X^2  0  0 X^3 X^3 X^3+X^2 X^3+X^2 X^2 X^2 X^3  0  0 X^3 X^3  0  0 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2
 0  0 X^3  0  0  0 X^3  0 X^3  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0  0 X^3  0  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0  0  0  0 X^3 X^3  0  0  0 X^3 X^3
 0  0  0 X^3  0  0  0  0  0  0  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0 X^3 X^3
 0  0  0  0 X^3  0 X^3 X^3 X^3  0  0  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3
 0  0  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0 X^3  0 X^3  0  0  0  0  0  0 X^3  0 X^3  0  0 X^3 X^3 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+67x^40+106x^42+623x^44+128x^45+36x^46+43x^48+18x^50+1x^52+1x^80

The gray image is a linear code over GF(2) with n=352, k=10 and d=160.
This code was found by Heurico 1.16 in 0.078 seconds.