The generator matrix

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

generates a code of length 43 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 37.

Homogenous weight enumerator: w(x)=1x^0+286x^37+1156x^38+3014x^39+4845x^40+7826x^41+9802x^42+11776x^43+9780x^44+8106x^45+4737x^46+2590x^47+1032x^48+390x^49+110x^50+44x^51+22x^52+16x^53+1x^54+2x^58

The gray image is a code over GF(2) with n=344, k=16 and d=148.
This code was found by Heurico 1.16 in 22.2 seconds.