The generator matrix

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

generates a code of length 43 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 39.

Homogenous weight enumerator: w(x)=1x^0+304x^39+784x^40+1216x^41+1374x^42+1308x^43+1221x^44+836x^45+546x^46+324x^47+145x^48+92x^49+22x^50+16x^51+1x^52+2x^54

The gray image is a code over GF(2) with n=344, k=13 and d=156.
This code was found by Heurico 1.16 in 0.407 seconds.