The generator matrix

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

generates a code of length 16 over Z3[X]/(X^3) who´s minimum homogenous weight is 21.

Homogenous weight enumerator: w(x)=1x^0+24x^21+80x^24+328x^27+428x^30+4374x^32+776x^33+446x^36+80x^39+8x^42+8x^45+8x^48

The gray image is a linear code over GF(3) with n=144, k=8 and d=63.
This code was found by Heurico 1.16 in 0.0449 seconds.