The generator matrix
1 0 1 1 1 0 1 1 X 1 1 X 1 1 1 1 0 X 1 1 1 2X 1 2X 1 1
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 1 X+1 1 2X 1
0 0 2X X X 2X 2X 0 X X 2X 2X 2X 0 0 X X 0 0 X 2X 0 X 2X 2X 0
generates a code of length 26 over Z3[X]/(X^2) who´s minimum homogenous weight is 51.
Homogenous weight enumerator: w(x)=1x^0+52x^51+162x^52+26x^54+2x^78
The gray image is a linear code over GF(3) with n=78, k=5 and d=51.
As d=51 is an upper bound for linear (78,5,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 15.6 seconds.