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 X^3  1  1  1  X  1  X  1 X^2  X  1 X^3+X^2  1  1  X  X  1  1
 0  X  0  X X^3  0 X^2+X X^2+X X^2 X^2 X^3+X^2+X  X X^3+X X^3+X^2 X^2 X^2+X X^2+X  X X^3 X^3 X^3+X^2+X X^3  X X^2  X X^2 X^3+X X^2  0  0  X  X X^3+X^2+X  0  X X^3+X^2  X X^3+X X^3+X^2+X X^3
 0  0  X  X X^2 X^2+X X^2+X  0 X^3+X^2  X X^2 X^3+X^2+X X^2 X^3 X^2+X X^3+X X^2+X  X X^2 X^3+X^2+X X^3+X^2 X^3 X^3+X X^3+X X^2 X^3+X^2+X  0 X^3 X^3+X  X X^3+X^2 X^3+X^2  X  X  0 X^3+X  0 X^2+X X^3+X X^3+X^2
 0  0  0 X^3 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3  0  0  0  0 X^3 X^3 X^3 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+73x^36+162x^37+337x^38+228x^39+525x^40+272x^41+238x^42+64x^43+63x^44+26x^45+41x^46+12x^47+1x^48+4x^49+1x^64

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 0.094 seconds.