The generator matrix
1 0 0 0 0 1 1 1 1 1 1 1 1
0 1 0 0 0 1 0 2X+1 2X+2 2X 1 2X+1 0
0 0 1 0 0 2 1 2X+1 2X X X+2 X+1 0
0 0 0 1 0 2 2 2X+1 2X+1 2X X+1 0 0
0 0 0 0 1 2 X+1 2X X+2 1 X+2 2X 0
0 0 0 0 0 2X X 0 2X 2X X X X
generates a code of length 13 over Z3[X]/(X^2) who´s minimum homogenous weight is 15.
Homogenous weight enumerator: w(x)=1x^0+222x^15+198x^16+348x^17+1398x^18+1392x^19+2760x^20+7314x^21+7476x^22+11400x^23+20478x^24+19608x^25+21576x^26+28184x^27+19398x^28+14844x^29+13044x^30+4296x^31+1560x^32+1494x^33+120x^34+36x^36
The gray image is a linear code over GF(3) with n=39, k=11 and d=15.
This code was found by Heurico 1.16 in 35 seconds.