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 2X  1  1  1  1  1  1  1  1  0  1  1  1  X  1  2  1  1  1
 0  X  0 X+2  2 3X+2 2X+2  X  0 X+2 2X+2 3X X+2 2X  X 2X+2 2X X+2  2 3X  0 3X+2  X 2X+2 2X 3X+2 2X+2  X 2X+2  X 2X 3X+2  0 X+2  0 X+2  2  X 2X+2 3X  0 X+2  0 X+2  2  X  2  X 2X+2 3X 2X+2 3X 2X+2 3X  2  X  0 2X X+2 3X+2 2X  0 3X+2 X+2 2X  0 2X 2X  0 X+2 3X+2 3X+2 X+2  X  0 3X+2 2X+2 3X+2 3X+2  2  X 3X 3X
 0  0 2X+2  0  2  2  0  2  0  0  0  0 2X+2 2X+2 2X+2  2  2 2X 2X+2 2X 2X  2 2X+2 2X 2X 2X 2X 2X 2X+2  2  2 2X+2  0  0 2X+2 2X+2  2 2X+2 2X  0  0  0 2X+2 2X+2 2X+2 2X  0  2 2X+2  2  0 2X  2 2X+2 2X  0 2X  2 2X  2 2X  2 2X  2  0 2X 2X  2  2 2X  2 2X  2 2X  2  0 2X 2X+2 2X+2 2X+2 2X+2  2  2
 0  0  0 2X  0  0  0 2X 2X 2X 2X  0  0 2X 2X 2X  0 2X  0  0 2X  0 2X 2X  0 2X  0  0 2X 2X 2X  0  0  0  0 2X 2X  0 2X 2X 2X  0 2X 2X  0 2X 2X  0 2X  0  0 2X  0  0  0 2X 2X  0  0 2X  0 2X  0 2X  0  0 2X 2X  0 2X 2X  0  0 2X  0  0  0 2X  0 2X  0 2X 2X
 0  0  0  0 2X  0 2X  0 2X 2X  0 2X 2X 2X 2X  0  0  0 2X 2X 2X 2X  0  0  0 2X 2X  0  0 2X 2X  0 2X  0 2X  0 2X  0 2X  0  0 2X  0 2X  0  0 2X 2X 2X  0  0 2X  0 2X  0 2X  0 2X 2X  0 2X  0  0 2X  0 2X 2X  0  0 2X 2X 2X 2X  0 2X  0  0 2X 2X 2X 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+28x^78+92x^79+137x^80+252x^81+443x^82+344x^83+304x^84+176x^85+63x^86+100x^87+38x^88+52x^89+9x^90+8x^91+1x^158

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 0.953 seconds.