The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+270x^28+1446x^29+4287x^30+8936x^31+14773x^32+23078x^33+25130x^34+23382x^35+15389x^36+8810x^37+3709x^38+1324x^39+410x^40+90x^41+24x^42+6x^43+3x^44+2x^46+2x^48

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 59.6 seconds.