The generator matrix

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

generates a code of length 16 over Z3[X]/(X^3) who´s minimum homogenous weight is 24.

Homogenous weight enumerator: w(x)=1x^0+380x^24+312x^25+3966x^26+4654x^27+16476x^28+20166x^29+69966x^30+63432x^31+118422x^32+133376x^33+63048x^34+27966x^35+8024x^36+1056x^37+66x^38+108x^39+18x^40+2x^42+2x^45

The gray image is a linear code over GF(3) with n=144, k=12 and d=72.
This code was found by Heurico 1.16 in 87.5 seconds.