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

generates a code of length 81 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+30x^78+264x^79+101x^80+288x^81+88x^82+176x^83+24x^84+32x^85+9x^86+8x^87+2x^88+1x^134

The gray image is a code over GF(2) with n=648, k=10 and d=312.
This code was found by Heurico 1.16 in 0.375 seconds.