The generator matrix

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

generates a code of length 40 over Z2[X]/(X^4) who´s minimum homogenous weight is 35.

Homogenous weight enumerator: w(x)=1x^0+114x^35+195x^36+344x^37+575x^38+448x^39+825x^40+474x^41+517x^42+286x^43+145x^44+100x^45+19x^46+12x^47+18x^48+10x^49+7x^50+4x^51+2x^54

The gray image is a linear code over GF(2) with n=320, k=12 and d=140.
This code was found by Heurico 1.16 in 56.5 seconds.