The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X X^3  1  1  X  1  1  1  1  1  1  1  1  1  1  1  1  0  X  X  1  1
 0  X  0 X^2+X X^2 X^3+X^2+X X^3+X^2  X  0 X^3+X X^3+X^2 X^2+X X^3 X^3+X^2+X  0 X^2+X X^3+X^2  X X^3+X^2+X  X  0 X^3+X X^3 X^2  X X^3  0 X^3+X X^2+X  X  X X^2+X X^2 X^2 X^2  X X^2  X  0 X^3+X^2
 0  0 X^3+X^2  0 X^2 X^2 X^3 X^2  0 X^3  0  0 X^2 X^2 X^3+X^2 X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3 X^3 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3 X^3+X^2 X^3  0 X^2 X^3+X^2  0 X^3+X^2 X^2  0  0
 0  0  0 X^3  0  0 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0
 0  0  0  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0 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+142x^36+56x^37+192x^38+480x^39+388x^40+432x^41+180x^42+32x^43+72x^44+24x^45+40x^46+3x^48+4x^50+1x^52+1x^68

The gray image is a linear code over GF(2) with n=320, k=11 and d=144.
This code was found by Heurico 1.16 in 0.11 seconds.