The generator matrix

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

generates a code of length 40 over Z2[X]/(X^4) who´s minimum homogenous weight is 37.

Homogenous weight enumerator: w(x)=1x^0+138x^37+485x^38+362x^39+156x^40+334x^41+418x^42+106x^43+8x^44+8x^45+17x^46+12x^47+1x^48+1x^52+1x^60

The gray image is a linear code over GF(2) with n=320, k=11 and d=148.
This code was found by Heurico 1.16 in 0.36 seconds.