The generator matrix

 1  0  1  1  1 X^2+X  1  1  1 X^3+X^2  1 X^3+X  1  1  1  0  1 X^2+X  1  1  1  1  0  1 X^2+X  1 X^3+X^2 X^3+X  1  1  1  1 X^3+X^2  1  1 X^3+X  1  1  1  1
 0  1 X+1 X^2+X X^2+1  1 X^3+X^2+X+1 X^3+1 X^3+X^2  1 X^3+X  1 X^2+1 X+1  0  1 X^2+X  1 X^3+X^2+X+1 X^3+1 X^3+X^2+X+1 X^3+X^2  1 X^2+X  1 X^3+X  1  1  0 X^2+1 X^3+X^2 X^2+1  1 X^3+X^2+1 X^3+X  1 X^3+1 X+1 X^2+X X^3+X^2+X
 0  0 X^3  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0  0 X^3
 0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3 X^3  0  0 X^3  0  0 X^3  0 X^3  0  0
 0  0  0  0 X^3  0 X^3 X^3  0  0  0  0 X^3 X^3  0  0  0  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3  0  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+50x^36+224x^37+154x^38+544x^39+107x^40+544x^41+148x^42+224x^43+44x^44+2x^46+3x^48+2x^52+1x^56

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 0.047 seconds.