The generator matrix

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

generates a code of length 34 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 29.

Homogenous weight enumerator: w(x)=1x^0+248x^29+1311x^30+2064x^31+4124x^32+5146x^33+7045x^34+5202x^35+4231x^36+1852x^37+1105x^38+276x^39+91x^40+50x^41+11x^42+10x^43+1x^44

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 4.41 seconds.