The generator matrix

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

generates a code of length 34 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 30.

Homogenous weight enumerator: w(x)=1x^0+654x^30+1192x^31+2020x^32+2944x^33+3056x^34+2896x^35+1711x^36+1088x^37+646x^38+72x^39+75x^40+28x^42+1x^44

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 59.4 seconds.