The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+152x^31+468x^32+268x^33+416x^34+204x^35+319x^36+108x^37+64x^38+28x^39+11x^40+8x^41+1x^44

The gray image is a code over GF(2) with n=272, k=11 and d=124.
This code was found by Heurico 1.16 in 0.032 seconds.