The generator matrix

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

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+80x^36+474x^37+520x^38+724x^39+647x^40+692x^41+439x^42+332x^43+71x^44+62x^45+27x^46+16x^47+4x^49+5x^50+1x^52+1x^54

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 0.14 seconds.