The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+297x^28+1420x^29+4251x^30+9208x^31+14367x^32+22810x^33+25669x^34+23690x^35+14973x^36+8702x^37+3786x^38+1396x^39+374x^40+88x^41+21x^42+10x^43+4x^44+2x^45+1x^46+2x^49

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 64.6 seconds.