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

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+57x^78+64x^79+198x^80+256x^81+306x^82+432x^83+191x^84+272x^85+66x^86+80x^87+57x^88+48x^89+18x^90+1x^94+1x^156

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 0.782 seconds.