The generator matrix

 1  0  1  1  1 X^3+X^2+X  1  1  X  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3  1  1  1 X^2  1 X^3+X  1  1  1  1  1  1  1  1  1  1  1  1  0  1  1 X^3+X  1  1  1  1
 0  1 X+1 X^2+X X^2+1  1 X^3+X^2+X X^2+X+1  1  X X+1  1 X^3+X^2 X^3+1  1 X^3 X^3+X^2+1  1 X^3+X X^2 X^3+X^2+X+1  1  1  1  0 X^3+X^2+X+1 X^3+X X^2+1  0 X^2+X X^2 X^3+X X^3+X  X X^3+X^2 X^2  1 X^2+X X^2+X  1 X^2 X^3+X^2  0  0
 0  0 X^2  0 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^2 X^3  0  0 X^3  0 X^2 X^3+X^2 X^2 X^3 X^2  0 X^2 X^3+X^2 X^3 X^3  0 X^3  0 X^2 X^3 X^3+X^2 X^3+X^2  0 X^2  0 X^3 X^3 X^3+X^2 X^2 X^3 X^2 X^3+X^2  0  0
 0  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0  0  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3  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+46x^40+248x^41+220x^42+328x^43+388x^44+328x^45+184x^46+248x^47+41x^48+12x^50+2x^52+2x^68

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 0.093 seconds.