The generator matrix

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

generates a code of length 34 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 32.

Homogenous weight enumerator: w(x)=1x^0+327x^32+384x^34+304x^36+4x^40+4x^48

The gray image is a code over GF(2) with n=272, k=10 and d=128.
This code was found by Heurico 1.16 in 0.281 seconds.