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

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+246x^32+576x^34+176x^36+24x^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 94.8 seconds.