The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+574x^32+384x^36+64x^40+1x^64

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 0.032 seconds.