The generator matrix

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

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+60x^30+190x^31+199x^32+468x^33+288x^34+452x^35+140x^36+136x^37+51x^38+30x^39+28x^40+4x^41+1x^54

The gray image is a code over GF(2) with n=272, k=11 and d=120.
This code was found by Heurico 1.16 in 0.063 seconds.