The generator matrix
1 0 1 1 1 0 1 1 X 1 1 X 1 1 1 1 0 X 1 1 1 1 2X 2X 1 1 2X
0 1 1 0 2X+1 1 X 2X+1 1 X X+1 1 0 X X+1 1 1 1 2X 2X 2X+1 X+1 1 1 2X 1 1
0 0 2X X X 2X 2X 0 X X 2X 2X 2X 0 0 X X 0 0 X 2X X 0 2X 2X 0 X
generates a code of length 27 over Z3[X]/(X^2) who´s minimum homogenous weight is 54.
Homogenous weight enumerator: w(x)=1x^0+240x^54+2x^81
The gray image is a linear code over GF(3) with n=81, k=5 and d=54.
As d=54 is an upper bound for linear (81,5,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 14.9 seconds.