The generator matrix

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

generates a code of length 40 over Z2[X]/(X^4) who´s minimum homogenous weight is 36.

Homogenous weight enumerator: w(x)=1x^0+403x^36+64x^37+620x^38+448x^39+1148x^40+448x^41+488x^42+64x^43+330x^44+44x^46+34x^48+3x^52+1x^64

The gray image is a linear code over GF(2) with n=320, k=12 and d=144.
This code was found by Heurico 1.16 in 103 seconds.