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  X  1  1  1  1  1  1  X  1  1  1  X  1  1 2X+2  1  1 2X+2  1 2X  0  1  1  X  1  1  1  1 2X  1  X  1  1  X  0  X  1 2X+2 2X  X  2
 0  X  0  X 2X  0 X+2 3X+2  0 2X  X 3X 2X X+2 2X X+2 2X  X 2X+2 3X+2  2  X 2X+2 3X X+2 2X+2  X 2X X+2  0 X+2 2X+2 2X+2 3X 3X+2 2X+2  X  X  2  0 3X  2 2X  0 X+2 X+2 3X+2  2 2X+2 X+2 2X+2 3X+2  2 X+2 3X+2 3X  X  X  X  X  0  X  X X+2  X 2X 3X  X 2X+2  0  X  2  0 3X+2 3X  X  X 3X+2 3X+2  2 2X 2X+2 2X+2
 0  0  X  X  0 3X+2 X+2 2X  2 3X+2 3X+2 2X+2 2X+2  X 3X 2X+2 2X 3X  X 2X+2  0 3X  X  2 3X+2  2  2 3X 2X  2 3X+2  X 3X+2  0 X+2  0 2X 3X 3X+2  0 X+2 2X 3X+2 3X+2 2X  X  2  2 3X+2 3X X+2  X 2X+2  0  2 2X  0 3X+2 2X+2  2 X+2 X+2 X+2 2X 3X+2 3X  X  0  2 2X+2  2 3X X+2 3X+2  X 3X 2X 3X+2  X  X  X  X  X
 0  0  0  2  2 2X+2  0 2X+2  2 2X  2 2X  0  0  2  2 2X 2X+2  2 2X  2  0  0  2 2X+2 2X 2X+2 2X  0 2X+2 2X 2X+2  2  2  2 2X 2X 2X 2X 2X+2 2X  0  0  2  2  2  0 2X+2  0  2 2X+2 2X  0 2X 2X+2 2X+2 2X+2 2X+2  0  2  0 2X+2  0 2X  0  2 2X+2 2X 2X+2  2  0  0 2X+2 2X+2  2  2 2X  0  2 2X 2X  2  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+352x^78+56x^79+611x^80+344x^81+582x^82+512x^83+477x^84+288x^85+400x^86+72x^87+232x^88+8x^89+74x^90+54x^92+28x^94+4x^98+1x^132

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 49.8 seconds.