The generator matrix

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

generates a code of length 86 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+142x^79+900x^80+1576x^81+2236x^82+2886x^83+3324x^84+3934x^85+3579x^86+3900x^87+3211x^88+2438x^89+1842x^90+1266x^91+686x^92+370x^93+228x^94+66x^95+80x^96+46x^97+26x^98+12x^99+5x^100+9x^102+4x^105+1x^108

The gray image is a code over GF(2) with n=688, k=15 and d=316.
This code was found by Heurico 1.16 in 15 seconds.