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
0 X 0 X 0 X 2X 2X 2X 0 0 X X 0 X 2X 2X 2X 0 0 X X 0 X 2X 2X 2X
0 0 X 2X 2X X 0 X 2X 0 X 2X X 2X 0 0 X 2X 0 X 2X X 2X 0 0 X 2X
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+78x^54+2x^81
The gray image is a linear code over GF(3) with n=81, k=4 and d=54.
As d=54 is an upper bound for linear (81,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.00202 seconds.