The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+106x^30+296x^31+664x^32+628x^33+796x^34+580x^35+621x^36+260x^37+91x^38+20x^39+15x^40+8x^41+6x^42+3x^44+1x^46

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 0.125 seconds.