The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+742x^29+2184x^30+4424x^31+8312x^32+10670x^33+12690x^34+11068x^35+8448x^36+4202x^37+1848x^38+706x^39+173x^40+50x^41+6x^42+8x^43+2x^44+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 16.7 seconds.