The generator matrix

 1  0  1  1  1 X^3+X^2+X  1  X  1 X^3+X^2  1  1  1  1 X^3  1  1 X^3+X^2+X  1  1 X^2+X  1 X^2  1  1  1 X^3+X  0 X^2+X  1  1 X^3  1  0 X^2  1  1  1  1  1
 0  1 X+1 X^2+X X^3+X^2+1  1 X^3+X^2  1 X^2+X+1  1 X^3+X^2+X X^2+1  X X^3+1  1 X^3+X+1  0  1 X^3+X  1  1 X^3  1 X^2+1 X+1 X^3+X^2+X+1  1  1  1  1 X^2  1 X^2+X  1  1 X^3+X^2+1 X^2+1 X^3+X^2+X+1 X^3+X^2 X^2+X
 0  0 X^2  0 X^3+X^2 X^2  0 X^2 X^3+X^2 X^3 X^2  0 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3 X^3+X^2 X^3 X^2  0 X^2 X^2  0 X^3 X^3 X^3 X^3 X^3 X^2 X^2 X^3+X^2 X^3 X^3+X^2 X^3+X^2  0 X^3  0  0  0
 0  0  0 X^3  0  0  0  0 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0  0  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3
 0  0  0  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0  0 X^3  0 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+208x^36+224x^37+698x^38+544x^39+781x^40+544x^41+668x^42+224x^43+180x^44+10x^46+5x^48+8x^52+1x^56

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 0.157 seconds.