The generator matrix

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

generates a code of length 81 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+130x^77+678x^78+686x^79+762x^80+388x^81+434x^82+230x^83+262x^84+106x^85+140x^86+120x^87+84x^88+36x^89+34x^90+3x^92+1x^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 0.39 seconds.