The generator matrix

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

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+402x^37+750x^38+828x^39+595x^40+532x^41+409x^42+300x^43+138x^44+102x^45+24x^46+8x^47+2x^48+4x^49+1x^50

The gray image is a linear code over GF(2) with n=320, k=12 and d=148.
This code was found by Heurico 1.16 in 0.109 seconds.