The generator matrix

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

generates a code of length 34 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 31.

Homogenous weight enumerator: w(x)=1x^0+296x^31+768x^32+760x^33+845x^34+494x^35+475x^36+202x^37+146x^38+90x^39+3x^40+14x^41+1x^42+1x^44

The gray image is a code over GF(2) with n=272, k=12 and d=124.
This code was found by Heurico 1.16 in 0.078 seconds.