The generator matrix

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

generates a code of length 81 over Z3[X]/(X^2) who´s minimum homogenous weight is 162.

Homogenous weight enumerator: w(x)=1x^0+726x^162+2x^243

The gray image is a linear code over GF(3) with n=243, k=6 and d=162.
As d=162 is an upper bound for linear (243,6,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 6.
This code was found by Heurico 1.16 in 0.115 seconds.