The generator matrix
1 1 1 1 1 1 1 1
0 X 2X X^2 2X^2+X X^2+2X 2X^2 X^2+X
generates a code of length 8 over Z3[X]/(X^3) who´s minimum homogenous weight is 15.
Homogenous weight enumerator: w(x)=1x^0+16x^15+216x^16+8x^18+2x^24
The gray image is a linear code over GF(3) with n=72, k=5 and d=45.
As d=46 is an upper bound for linear (72,5,3)-codes, this code is optimal over Z3[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 0.000181 seconds.