The generator matrix

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

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+480x^40+640x^41+804x^42+744x^43+491x^44+328x^45+312x^46+120x^47+113x^48+24x^49+28x^50+11x^52

The gray image is a code over GF(2) with n=344, k=12 and d=160.
This code was found by Heurico 1.16 in 0.297 seconds.