The generator matrix

 1  0  1  1  1 X^2+X+2  1  1 X^2+2  1  1  2  1  1  X  1  1  1  1  X  1  1 X^2 X^2+X X^2+2 X+2  2 X^2+X  X  2 X^2+X+2  2 X^2+X+2  0  1  1  1  1  1  1  0 X^2+2 X+2
 0  1 X+1 X^2+X+2 X^2+1  1  2 X^2+X+1  1 X^2+2 X+1  1  X  3  1 X^2 X^2+3 X^2+X X^2+X+3  1 X+2  1  1  1  1  1  1  1  1  1  1  1  1  1  2 X^2+X+3  1 X^2+2 X^2+1 X^2+X+3  0  1  1
 0  0 X^2 X^2  2 X^2 X^2+2 X^2+2  2  2  0 X^2+2 X^2  2 X^2 X^2+2 X^2  0  0 X^2  2 X^2+2 X^2+2  0  0  2 X^2  2  0  2 X^2+2 X^2+2  2 X^2 X^2  2 X^2+2  2  0  0 X^2  0  2
 0  0  0  2  0  0  0  2  2  2  2  2  0  2  2  2  2  2  0  0  0  0  2  0  2  0  0  2  2  0  2  0  0  2  2  2  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 40.

Homogenous weight enumerator: w(x)=1x^0+308x^40+304x^41+244x^42+336x^43+355x^44+208x^45+204x^46+48x^47+34x^48+4x^52+1x^60+1x^64

The gray image is a code over GF(2) with n=344, k=11 and d=160.
This code was found by Heurico 1.16 in 18.5 seconds.