The generator matrix
1 0 1 1 1 1 1 0 1 1 1 X 1 1 1 X 1 1 1 1 1 1 0 X 1 1 X 1
0 1 1 2 2X+1 0 2 1 X 2X+1 X+2 1 X X+1 X+2 1 X+1 1 0 X 2 X+2 1 1 2X 2X 0 2X
0 0 2X 0 X X 2X 2X 2X 0 X X X 2X 2X 2X X 0 2X 0 X 0 X 0 0 X 2X 0
generates a code of length 28 over Z3[X]/(X^2) who´s minimum homogenous weight is 54.
Homogenous weight enumerator: w(x)=1x^0+128x^54+72x^57+18x^60+24x^63
The gray image is a linear code over GF(3) with n=84, k=5 and d=54.
As d=54 is an upper bound for linear (84,5,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.00286 seconds.