The generator matrix

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

generates a code of length 34 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 28.

Homogenous weight enumerator: w(x)=1x^0+306x^28+1530x^29+4157x^30+8568x^31+15376x^32+22496x^33+25884x^34+22984x^35+15491x^36+8580x^37+3876x^38+1272x^39+407x^40+96x^41+32x^42+8x^43+3x^44+2x^45+3x^46

The gray image is a code over GF(2) with n=272, k=17 and d=112.
This code was found by Heurico 1.16 in 63 seconds.