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

generates a code of length 83 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+168x^79+642x^80+782x^81+664x^82+402x^83+318x^84+282x^85+253x^86+206x^87+129x^88+104x^89+72x^90+24x^91+29x^92+16x^93+1x^94+2x^98+1x^100

The gray image is a code over GF(2) with n=664, k=12 and d=316.
This code was found by Heurico 1.16 in 0.453 seconds.