The generator matrix

 1  1  1  1  1  1  1  1  1  1  X  X  1  1  1  X  X  0 X^2
 0  X 2X X^2 2X^2+X X^2+2X 2X^2 X^2+X 2X^2+2X  0 2X^2+X 2X 2X^2+X X^2 X^2+X X^2+X X^2+2X  X  X

generates a code of length 19 over Z3[X]/(X^3) who´s minimum homogenous weight is 37.

Homogenous weight enumerator: w(x)=1x^0+132x^37+36x^38+2x^39+66x^40+4x^42+2x^48

The gray image is a linear code over GF(3) with n=171, k=5 and d=111.
As d=113 is an upper bound for linear (171,5,3)-codes, this code is optimal over Z3[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 0.00178 seconds.