The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+232x^79+636x^80+716x^81+598x^82+416x^83+366x^84+284x^85+302x^86+184x^87+104x^88+80x^89+92x^90+72x^91+8x^92+1x^96+4x^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 0.406 seconds.