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  1  1  1  1  1  1  1 X^2  1 X^2  1  1
 0 X^3+X^2  0 X^2  0  0 X^2 X^3+X^2 X^3+X^2 X^3+X^2  0 X^3  0 X^3 X^3+X^2 X^3+X^2  0 X^2 X^2 X^3+X^2 X^2 X^3 X^3 X^3 X^3 X^3+X^2 X^3+X^2 X^2 X^3 X^3 X^2  0  0  0 X^3+X^2  0  0 X^3+X^2 X^2 X^3  0 X^2 X^3 X^3
 0  0 X^3+X^2 X^2  0 X^3+X^2 X^3+X^2  0 X^3 X^3+X^2  0 X^2  0 X^2 X^2 X^3 X^3 X^2 X^3 X^2  0 X^3 X^2 X^3+X^2  0  0 X^2 X^2 X^3+X^2 X^3  0 X^3+X^2 X^2 X^3  0 X^3 X^2 X^3  0 X^3 X^3 X^2 X^3 X^2
 0  0  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0  0 X^3  0  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  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3  0  0  0  0 X^3  0  0 X^3 X^3  0 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+51x^40+84x^42+192x^43+389x^44+192x^45+64x^46+32x^48+12x^50+6x^52+1x^84

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 0.063 seconds.