The generator matrix
1 1 1 1 1 1 1 1 1
0 X X^2 2X^2+X 2X^2 X^2+X 2X X^2+2X 2X^2+2X
generates a code of length 9 over Z3[X]/(X^3) who´s minimum homogenous weight is 18.
Homogenous weight enumerator: w(x)=1x^0+240x^18+2x^27
The gray image is a linear code over GF(3) with n=81, k=5 and d=54.
As d=54 is an upper bound for linear (81,5,3)-codes, this code is optimal over Z3[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 0.000206 seconds.