The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  1  1  1  1  1  1  1  1  X  1  1 X^2  1  X  1  1  0  1 X^2
 0 X^3+X^2  0  0  0 X^2 X^3+X^2 X^2  0 X^3 X^3+X^2 X^3+X^2  0 X^3 X^3+X^2 X^3+X^2  0 X^3 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^2 X^3  0 X^3 X^2 X^2 X^3+X^2 X^3  0  0 X^3 X^2 X^3+X^2  0 X^3+X^2 X^2 X^3+X^2 X^2
 0  0 X^3+X^2  0 X^2 X^2 X^2 X^3  0 X^3 X^2 X^3+X^2 X^2 X^2 X^3 X^3  0 X^3+X^2 X^3+X^2  0  0 X^2  0  0 X^2  0 X^3+X^2 X^3 X^2 X^3+X^2 X^3  0 X^2 X^3+X^2 X^3 X^3  0 X^3  0 X^2
 0  0  0 X^3+X^2 X^2 X^3 X^3+X^2 X^3+X^2  0 X^3+X^2 X^3 X^3+X^2 X^2  0 X^3+X^2  0 X^3 X^3+X^2 X^2 X^2 X^3+X^2  0  0  0 X^3 X^2 X^2 X^3 X^3 X^2  0 X^2 X^3  0 X^3  0 X^3+X^2 X^3 X^3+X^2  0
 0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3  0  0  0  0  0 X^3 X^3 X^3  0  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+199x^36+84x^37+704x^39+154x^40+648x^41+96x^43+112x^44+4x^45+45x^48+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 8.19 seconds.