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

generates a code of length 82 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+180x^78+64x^79+437x^80+128x^81+456x^82+320x^83+224x^84+132x^86+105x^88+1x^152

The gray image is a code over GF(2) with n=656, k=11 and d=312.
This code was found by Heurico 1.16 in 137 seconds.