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  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 X^2  1  X
 0 X^3+X^2  0  0 X^3 X^3+X^2 X^3+X^2 X^2  0 X^3  0 X^2 X^3+X^2 X^3 X^3+X^2 X^3+X^2  0 X^2 X^2  0 X^2 X^3 X^2  0 X^2  0 X^3 X^2  0 X^3 X^3+X^2 X^3 X^3 X^3 X^3 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0
 0  0 X^3+X^2  0 X^3+X^2 X^3+X^2 X^2 X^3  0 X^2 X^2 X^3+X^2 X^3+X^2 X^3  0  0 X^3 X^3+X^2 X^2 X^3+X^2  0 X^2  0 X^3 X^3+X^2 X^2 X^3 X^3  0 X^3+X^2 X^3+X^2  0 X^3 X^3+X^2 X^2  0 X^2 X^3 X^2 X^3
 0  0  0 X^3+X^2 X^2 X^3 X^2 X^2 X^3 X^3 X^2 X^3 X^3+X^2 X^2 X^3+X^2 X^3 X^3 X^2 X^3 X^2 X^3  0 X^2 X^3+X^2 X^3+X^2 X^3  0 X^3+X^2 X^2  0  0 X^3 X^3 X^3+X^2 X^2  0 X^3  0 X^3+X^2 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+55x^36+50x^38+192x^39+431x^40+192x^41+60x^42+24x^44+18x^46+1x^76

The gray image is a linear code over GF(2) with n=320, k=10 and d=144.
This code was found by Heurico 1.16 in 0.047 seconds.