The generator matrix

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

generates a code of length 12 over Z3[X]/(X^3) who´s minimum homogenous weight is 18.

Homogenous weight enumerator: w(x)=1x^0+420x^18+486x^20+1440x^21+1458x^22+1944x^23+6966x^24+2916x^25+1944x^26+1928x^27+180x^30

The gray image is a linear code over GF(3) with n=108, k=9 and d=54.
This code was found by Heurico 1.16 in 0.204 seconds.