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

generates a code of length 81 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+172x^77+73x^78+224x^79+190x^80+744x^81+191x^82+224x^83+31x^84+172x^85+23x^86+1x^88+1x^90+1x^156

The gray image is a code over GF(2) with n=648, k=11 and d=308.
This code was found by Heurico 1.16 in 1.11 seconds.