The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  0  1  1  1  1  1  1  1  1 X^3+X^2  1  1  1  1  1  1  X  0  1  1  X  1  X  1
 0  X  0 X^3+X^2+X X^3 X^2+X  0  X X^2 X^3+X^2+X X^3+X^2 X^3+X X^3+X^2 X^3+X^2+X X^3+X^2  X  X  X X^3 X^3+X X^3+X X^3 X^3+X^2 X^3+X^2+X  0  X  0 X^3+X X^3+X^2+X X^3+X  0  X X^3+X^2+X X^2 X^2+X  X X^2+X X^3+X^2+X X^2+X X^3
 0  0 X^3+X^2  0 X^3 X^3+X^2 X^3+X^2 X^2 X^2 X^2 X^3 X^2 X^2  0  0 X^3 X^3  0 X^2  0 X^2  0 X^3+X^2  0 X^2 X^3  0 X^3+X^2 X^2 X^3  0 X^3+X^2 X^3+X^2 X^3+X^2 X^3 X^3 X^3+X^2 X^3  0 X^3
 0  0  0 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^3  0  0 X^3+X^2 X^3+X^2 X^2  0 X^3 X^3+X^2 X^3 X^3+X^2 X^3 X^3  0 X^3+X^2 X^3+X^2 X^3  0 X^3 X^3 X^2 X^3+X^2 X^2 X^2  0  0 X^3+X^2 X^3  0 X^2 X^3  0 X^2

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+111x^36+124x^37+272x^38+312x^39+440x^40+368x^41+212x^42+72x^43+76x^44+20x^45+26x^46+11x^48+2x^54+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 0.344 seconds.