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

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+36x^78+40x^79+163x^80+248x^81+452x^82+352x^83+350x^84+160x^85+48x^86+56x^87+101x^88+40x^89+1x^160

The gray image is a code over GF(2) with n=664, k=11 and d=312.
This code was found by Heurico 1.16 in 0.719 seconds.