The generator matrix

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

generates a code of length 17 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 13.

Homogenous weight enumerator: w(x)=1x^0+550x^13+2276x^14+6512x^15+14093x^16+18430x^17+14562x^18+6440x^19+2048x^20+570x^21+40x^22+8x^23+2x^24+2x^25+2x^26

The gray image is a code over GF(2) with n=136, k=16 and d=52.
This code was found by Heurico 1.16 in 6.95 seconds.