The generator matrix
1 1 1
0 X^2 2X^2
generates a code of length 3 over Z3[X]/(X^3) who´s minimum homogenous weight is 6.
Homogenous weight enumerator: w(x)=1x^0+78x^6+2x^9
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^3) for dimension 4.
This code was found by Heurico 1.16 in 2.81e-005 seconds.