The generator matrix

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

generates a code of length 85 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+192x^79+767x^80+1368x^81+1674x^82+1910x^83+1952x^84+1854x^85+1680x^86+1380x^87+1135x^88+880x^89+669x^90+402x^91+228x^92+174x^93+64x^94+16x^95+8x^96+12x^97+8x^98+4x^99+4x^100+1x^104+1x^106

The gray image is a code over GF(2) with n=680, k=14 and d=316.
This code was found by Heurico 1.16 in 4.2 seconds.