The generator matrix

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

generates a code of length 34 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 30.

Homogenous weight enumerator: w(x)=1x^0+318x^30+16x^31+799x^32+368x^33+1232x^34+368x^35+627x^36+16x^37+258x^38+76x^40+12x^42+4x^46+1x^52

The gray image is a code over GF(2) with n=272, k=12 and d=120.
This code was found by Heurico 1.16 in 0.547 seconds.