The generator matrix

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

generates a code of length 83 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+256x^78+240x^79+500x^80+320x^81+535x^82+416x^83+533x^84+320x^85+470x^86+240x^87+241x^88+17x^90+1x^92+1x^94+1x^96+1x^102+2x^108+1x^112

The gray image is a code over GF(2) with n=664, k=12 and d=312.
This code was found by Heurico 1.16 in 0.781 seconds.