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

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+444x^40+756x^41+672x^42+800x^43+477x^44+384x^45+230x^46+144x^47+133x^48+28x^49+16x^50+9x^52+2x^54

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 0.125 seconds.