The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  0  1  2  2  1  2  1  1  1  X  X  X  1  X  1  X  X  1  1
 0  X  0 X^2+X+2  2 X^2+X  0  X X^2 X^2+X+2 X^2+2 X+2 X^2+2 X^2+2 X^2+X+2  X  X  X  X  2  X  2 X^2+X X^2+X X^2+X X^2+X  X X^2 X+2  0 X+2 X+2 X+2 X^2+X
 0  0 X^2+2  0  2 X^2+2 X^2+2 X^2 X^2 X^2  2 X^2 X^2  0  0  2  2 X^2+2  0  0 X^2+2 X^2  0 X^2+2  2 X^2+2  2 X^2+2 X^2+2  0 X^2  2 X^2+2 X^2+2
 0  0  0 X^2+2 X^2+2 X^2 X^2+2  2  0  0 X^2+2 X^2+2 X^2  2  2  2 X^2  2 X^2+2 X^2 X^2  2  0 X^2+2  0 X^2 X^2+2 X^2+2  2 X^2+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+96x^30+128x^31+399x^32+192x^33+456x^34+192x^35+356x^36+128x^37+68x^38+17x^40+12x^42+2x^44+1x^48

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 0.062 seconds.