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  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  1  1  1  1  1  1  1  1  1  X  X  X  X  X  X  X  X  X  X  X  X  1  1  1  1  1  X X^2  1  1
 0 X^2+2  0 X^2  0  0 X^2 X^2+2  0  0 X^2 X^2+2  0  0 X^2 X^2+2  2  2 X^2+2 X^2  2  2 X^2+2 X^2  2  2 X^2+2 X^2  2  2 X^2+2 X^2  2  0 X^2+2 X^2 X^2 X^2 X^2+2 X^2  2  0 X^2 X^2  0  0  2  2 X^2+2 X^2+2 X^2+2  0  2 X^2+2  2 X^2 X^2+2 X^2 X^2+2  0 X^2  2  0  2  0 X^2+2 X^2 X^2+2 X^2+2  2  0  0 X^2+2 X^2+2  0  2 X^2+2  0  2  2  0
 0  0 X^2+2 X^2  0 X^2+2 X^2  0  2 X^2 X^2+2  2  2 X^2 X^2+2  2  2 X^2 X^2+2  2  2 X^2 X^2+2  2  0 X^2+2 X^2  0  0 X^2+2 X^2  0 X^2 X^2+2  0  2 X^2 X^2+2  0  2 X^2 X^2+2 X^2 X^2+2  0  2  2  0 X^2+2 X^2  2  0  0  0 X^2 X^2 X^2 X^2+2 X^2+2 X^2+2  0 X^2+2 X^2 X^2+2 X^2  2  0 X^2+2 X^2  2  0  2  2  0  0  0  2  2  2 X^2+2 X^2
 0  0  0  2  2  2  0  2  2  2  2  2  0  0  0  0  0  0  0  0  2  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  2  2  2  2  2  2  0  0  2  2  2  2  0  0  2  0  0  2  2  2  2  2  2  2  2  2  2  0  0  0  0  2  2  0  0  0  2  0  2  2  0  2  2  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+172x^80+256x^81+64x^82+16x^88+1x^96+2x^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 38.7 seconds.