The generator matrix

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

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+23x^40+26x^42+64x^43+284x^44+64x^45+28x^46+11x^48+10x^50+1x^72

The gray image is a linear code over GF(2) with n=352, k=9 and d=160.
This code was found by Heurico 1.16 in 0.047 seconds.