The generator matrix

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

generates a code of length 86 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+474x^78+1824x^79+3424x^80+5646x^81+8486x^82+10702x^83+13041x^84+14208x^85+15673x^86+14652x^87+13510x^88+10672x^89+7718x^90+4994x^91+2954x^92+1588x^93+792x^94+394x^95+159x^96+74x^97+56x^98+8x^99+13x^100+4x^101+1x^102+2x^103+2x^104

The gray image is a code over GF(2) with n=688, k=17 and d=312.
This code was found by Heurico 1.16 in 220 seconds.