The generator matrix

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

generates a code of length 13 over Z3[X]/(X^2) who´s minimum homogenous weight is 24.

Homogenous weight enumerator: w(x)=1x^0+116x^24+90x^27+30x^30+4x^33+2x^36

The gray image is a linear code over GF(3) with n=39, k=5 and d=24.
As d=24 is an upper bound for linear (39,5,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.000585 seconds.