The generator matrix

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

generates a code of length 34 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 29.

Homogenous weight enumerator: w(x)=1x^0+692x^29+2258x^30+4482x^31+8241x^32+10588x^33+12802x^34+10986x^35+8420x^36+4302x^37+1870x^38+648x^39+186x^40+32x^41+14x^42+10x^43+2x^45+2x^47

The gray image is a code over GF(2) with n=272, k=16 and d=116.
This code was found by Heurico 1.16 in 17.7 seconds.