The generator matrix

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

generates a code of length 40 over Z2[X]/(X^4) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+114x^34+1120x^35+2731x^36+5088x^37+7144x^38+10490x^39+12087x^40+10534x^41+7758x^42+4900x^43+2085x^44+972x^45+308x^46+162x^47+24x^48+14x^49+4x^50

The gray image is a linear code over GF(2) with n=320, k=16 and d=136.
This code was found by Heurico 1.16 in 24.2 seconds.