The generator matrix

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

generates a code of length 34 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 27.

Homogenous weight enumerator: w(x)=1x^0+100x^27+904x^28+3102x^29+7414x^30+17336x^31+30403x^32+45692x^33+51144x^34+46696x^35+31370x^36+17236x^37+6886x^38+2632x^39+800x^40+300x^41+92x^42+20x^43+10x^44+6x^45

The gray image is a code over GF(2) with n=272, k=18 and d=108.
This code was found by Heurico 1.16 in 244 seconds.