The generator matrix

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

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+104x^77+762x^78+658x^79+654x^80+486x^81+390x^82+234x^83+282x^84+130x^85+148x^86+92x^87+77x^88+24x^89+49x^90+2x^92+2x^94+1x^98

The gray image is a code over GF(2) with n=648, k=12 and d=308.
This code was found by Heurico 1.16 in 0.422 seconds.