The generator matrix

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

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+178x^30+740x^31+1114x^32+1300x^33+1751x^34+1248x^35+938x^36+584x^37+188x^38+76x^39+43x^40+20x^41+11x^42

The gray image is a code over GF(2) with n=272, k=13 and d=120.
This code was found by Heurico 1.16 in 0.297 seconds.