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.