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

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+96x^41+236x^42+192x^43+208x^44+96x^45+48x^46+36x^48+4x^50+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 0.093 seconds.