The generator matrix

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

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+324x^30+88x^31+596x^32+688x^33+814x^34+672x^35+490x^36+80x^37+258x^38+8x^39+63x^40+10x^42+1x^44+2x^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 29.7 seconds.