The generator matrix

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

generates a code of length 6 over Z3[X]/(X^3) who´s minimum homogenous weight is 12.

Homogenous weight enumerator: w(x)=1x^0+78x^12+2x^18

The gray image is a linear code over GF(3) with n=54, k=4 and d=36.
As d=36 is an upper bound for linear (54,4,3)-codes, this code is optimal over Z3[X]/(X^3) for dimension 4.
This code was found by Heurico 1.16 in 7.49e-005 seconds.