The generator matrix

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

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+254x^29+1208x^30+2454x^31+4116x^32+5112x^33+6583x^34+5192x^35+4084x^36+2272x^37+1030x^38+302x^39+117x^40+24x^41+11x^42+4x^43+2x^44+2x^45

The gray image is a code over GF(2) with n=272, k=15 and d=116.
This code was found by Heurico 1.16 in 4.72 seconds.