The generator matrix

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

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+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 0.266 seconds.