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 X^2  1 X^3  1 X^2  X  1
 0  X  0 X^2+X X^2 X^3+X^2+X X^3+X^2  X X^3+X X^3  0 X^2+X X^3+X^2 X^3+X^2 X^3+X^2+X  X X^2+X X^2+X  0 X^3 X^3+X X^3+X X^3 X^2 X^3+X^2+X  X X^2 X^2 X^3 X^2+X X^2  0  X X^3 X^2+X X^3+X^2+X X^3+X^2+X X^3+X^2  X  X  X X^3+X^2 X^3+X^2+X X^3+X^2+X
 0  0 X^3+X^2  0 X^2 X^2  0 X^2 X^3 X^2  0  0 X^3 X^2 X^2 X^3+X^2 X^3 X^3+X^2  0 X^2  0 X^3+X^2 X^3 X^2 X^3 X^3 X^3+X^2  0 X^2 X^3+X^2 X^3 X^3 X^2  0 X^3 X^3+X^2 X^3 X^3 X^3 X^3+X^2 X^3 X^3 X^2 X^3+X^2
 0  0  0 X^3  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  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 X^3 X^3  0 X^3
 0  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0  0  0 X^3  0 X^3 X^3  0 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3  0 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+198x^40+96x^41+132x^42+352x^43+504x^44+288x^45+248x^46+32x^47+184x^48+4x^50+8x^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 0.14 seconds.