The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+676x^30+1112x^31+2322x^32+2432x^33+3406x^34+2608x^35+2075x^36+960x^37+626x^38+56x^39+81x^40+26x^42+1x^44+2x^46

The gray image is a code over GF(2) with n=272, k=14 and d=120.
This code was found by Heurico 1.16 in 42.5 seconds.