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  X  X  1  1  1  1  1  1  1  1  1  1
 0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2 X^2  0 X^3 X^3+X^2  0 X^3+X^2 X^3 X^2  0 X^3+X^2 X^3 X^2 X^3 X^2 X^3 X^2  0 X^3+X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3+X^2 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3  0 X^3+X^2 X^3  0
 0  0 X^3  0  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0 X^3 X^3  0  0  0  0  0  0  0 X^3 X^3  0 X^3  0 X^3  0  0
 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 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0  0  0  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3  0
 0  0  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3  0  0  0  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3 X^3 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+4x^40+16x^41+34x^42+16x^43+374x^44+16x^45+28x^46+16x^47+3x^48+2x^50+1x^52+1x^84

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 0.062 seconds.