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

generates a code of length 81 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+321x^80+128x^82+48x^84+12x^88+1x^96+1x^112

The gray image is a code over GF(2) with n=648, k=9 and d=320.
This code was found by Heurico 1.16 in 54.8 seconds.