The generator matrix

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

generates a code of length 43 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 42.

Homogenous weight enumerator: w(x)=1x^0+12x^42+96x^43+12x^44+2x^46+3x^48+2x^54

The gray image is a code over GF(2) with n=344, k=7 and d=168.
This code was found by Heurico 1.16 in 0.046 seconds.