The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+471x^78+40x^79+424x^80+256x^81+759x^82+432x^83+481x^84+256x^85+589x^86+40x^87+214x^88+85x^90+19x^92+16x^94+12x^96+1x^136

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 17 seconds.