The generator matrix

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

generates a code of length 38 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+158x^34+228x^35+789x^36+430x^37+952x^38+402x^39+758x^40+204x^41+134x^42+8x^43+9x^44+6x^45+10x^46+2x^47+3x^48+2x^54

The gray image is a code over GF(2) with n=304, k=12 and d=136.
This code was found by Heurico 1.16 in 8.47 seconds.