The generator matrix

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

generates a code of length 86 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+192x^82+752x^83+680x^84+556x^85+398x^86+408x^87+296x^88+244x^89+142x^90+168x^91+84x^92+88x^93+50x^94+24x^95+9x^96+2x^100+2x^102

The gray image is a code over GF(2) with n=688, k=12 and d=328.
This code was found by Heurico 1.16 in 0.5 seconds.