The generator matrix
1 1 1 1 1 1 1 1 1
0 X 0 X 0 X 2X 2X 2X
0 0 X 2X 2X X 0 X 2X
generates a code of length 9 over Z3[X]/(X^2) who´s minimum homogenous weight is 18.
Homogenous weight enumerator: w(x)=1x^0+78x^18+2x^27
The gray image is a linear code over GF(3) with n=27, k=4 and d=18.
As d=18 is an upper bound for linear (27,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.000151 seconds.