The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X X 1 1 1 X X X X 1 1 1
0 X^2 2X^2 0 X^2 2X^2 0 X^2 2X^2 0 X^2 2X^2 0 X^2 2X^2 X^2 2X^2 0 X^2 2X^2 0 X^2 2X^2 0 0 X^2 2X^2
generates a code of length 27 over Z3[X]/(X^3) who´s minimum homogenous weight is 54.
Homogenous weight enumerator: w(x)=1x^0+72x^54+6x^60+2x^63
The gray image is a linear code over GF(3) with n=243, k=4 and d=162.
As d=162 is an upper bound for linear (243,4,3)-codes, this code is optimal over Z3[X]/(X^3) for dimension 4.
This code was found by Heurico 1.16 in 0.0033 seconds.