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.