The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 X^3  1  1  1  1  1  1  1  1 X^2  1  1  1  1  1  1  1  X  1  1  X  1  X  1  1  X  1
 0  X  0 X^3+X^2+X X^3 X^2+X  0  X X^2 X^3+X^2+X X^3+X^2 X^3+X X^3+X^2 X^3+X^2+X X^3+X^2  X X^3+X^2  X X^2+X  0  0  X X^2+X X^3+X X^3 X^2+X  X X^3+X X^2 X^2+X X^2  X X^2+X X^3+X^2 X^2+X X^3+X X^3+X^2 X^3+X^2+X  0 X^2+X X^2+X  X  X X^2
 0  0 X^3+X^2  0 X^3 X^3+X^2 X^3+X^2 X^2 X^2 X^2 X^3 X^2 X^2  0  0 X^3 X^3+X^2  0 X^3+X^2 X^3 X^2 X^3+X^2  0  0 X^3 X^2  0  0 X^3 X^3 X^3 X^3+X^2 X^2 X^3+X^2  0  0 X^3+X^2  0 X^2 X^2 X^3 X^2 X^2 X^3+X^2
 0  0  0 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^3  0  0 X^3+X^2 X^3+X^2 X^2  0 X^3 X^3+X^2 X^2 X^3+X^2 X^3+X^2 X^3 X^3 X^3 X^3 X^2 X^2 X^2 X^3+X^2 X^3  0 X^2 X^2 X^2 X^3 X^3+X^2  0 X^3+X^2  0 X^2 X^2 X^2  0  0 X^3 X^3+X^2

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+179x^40+32x^41+210x^42+480x^43+266x^44+480x^45+204x^46+32x^47+144x^48+2x^50+13x^52+4x^56+1x^76

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 0.157 seconds.