The generator matrix

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

generates a code of length 43 over Z2[X]/(X^4) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+106x^40+128x^42+512x^43+248x^44+12x^48+16x^52+1x^80

The gray image is a linear code over GF(2) with n=344, k=10 and d=160.
This code was found by Heurico 1.16 in 6.11 seconds.