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  X  1  1  1  1  1  1  1  1
 0  X  0 X^2+X X^2 X^3+X^2+X X^3+X^2  X  0 X^2+X X^3+X^2 X^3+X X^3  X X^3+X^2 X^3+X^2+X  X  0 X^3+X^2 X^2+X X^2+X X^3 X^2 X^3+X X^3+X  0 X^2+X X^2+X  0 X^3+X^2 X^3+X^2 X^3+X X^3+X X^3  0 X^3 X^2+X X^3+X^2+X X^3+X^2+X X^3 X^2 X^3+X^2 X^3
 0  0 X^3+X^2  0 X^3+X^2 X^2  0 X^2 X^3 X^3 X^3 X^3 X^2 X^3+X^2 X^2 X^3+X^2 X^2  0  0  0 X^3+X^2 X^2 X^2 X^3 X^3+X^2 X^3 X^2 X^3 X^3+X^2 X^3+X^2 X^3  0 X^3+X^2 X^3  0  0 X^3+X^2 X^3  0 X^3+X^2 X^3 X^3+X^2 X^3
 0  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3  0  0  0 X^3 X^3 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+105x^40+96x^42+640x^43+91x^44+64x^46+26x^48+1x^84

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 0.078 seconds.