The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 X 2X X^2 2X^2+X X^2+2X 2X^2 X^2+X 2X^2+2X 0 2X^2+X 2X X^2 X^2+X X^2+2X
generates a code of length 15 over Z3[X]/(X^3) who´s minimum homogenous weight is 29.
Homogenous weight enumerator: w(x)=1x^0+36x^29+186x^30+18x^32+2x^45
The gray image is a linear code over GF(3) with n=135, k=5 and d=87.
As d=89 is an upper bound for linear (135,5,3)-codes, this code is optimal over Z3[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 0.00165 seconds.