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 2X  X 2X+2
 0  X  2 3X+2  0 3X+2  2 3X  0 3X+2 3X  2 2X X+2 2X+2 3X  0 3X+2  2 3X  0 3X+2  2 3X  2 3X+2  0 3X 2X X+2 2X+2  X  0 3X+2  2  X 2X 2X 3X X+2 2X  0  2
 0  0 2X  0  0  0 2X  0  0 2X 2X 2X  0 2X 2X 2X 2X  0  0 2X 2X 2X  0  0  0  0 2X  0 2X 2X  0 2X  0 2X 2X 2X 2X  0 2X  0 2X  0  0
 0  0  0 2X  0  0  0 2X 2X 2X 2X 2X 2X  0 2X  0 2X  0  0 2X 2X 2X 2X 2X  0 2X  0  0  0  0 2X  0 2X  0 2X  0  0  0  0  0  0 2X 2X
 0  0  0  0 2X 2X 2X 2X 2X  0 2X  0  0 2X 2X  0  0  0  0  0 2X 2X 2X  0 2X 2X 2X 2X  0  0  0 2X  0 2X 2X  0 2X  0 2X 2X 2X 2X 2X

generates a code of length 43 over Z4[X]/(X^2+2X+2) 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 code over GF(2) with n=344, k=10 and d=160.
This code was found by Heurico 1.16 in 0.093 seconds.