The generator matrix

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

generates a code of length 13 over Z3[X]/(X^2) who´s minimum homogenous weight is 20.

Homogenous weight enumerator: w(x)=1x^0+390x^20+260x^21+780x^23+702x^24+1800x^26+728x^27+1248x^29+494x^30+156x^32+2x^39

The gray image is a linear code over GF(3) with n=39, k=8 and d=20.
This code was found by Heurico 1.16 in 575 seconds.