The generator matrix

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

generates a code of length 12 over Z3[X]/(X^2) who´s minimum homogenous weight is 12.

Homogenous weight enumerator: w(x)=1x^0+42x^12+54x^14+204x^15+162x^16+540x^17+1392x^18+1296x^19+2160x^20+6198x^21+3888x^22+4320x^23+16098x^24+5184x^25+4320x^26+8120x^27+2592x^28+1728x^29+618x^30+114x^33+18x^36

The gray image is a linear code over GF(3) with n=36, k=10 and d=12.
This code was found by Heurico 1.16 in 1.05 seconds.