The generator matrix

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

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+32x^24+156x^26+92x^27+810x^28+618x^29+1662x^30+3240x^31+5634x^32+3128x^33+3240x^34+834x^35+132x^36+48x^38+48x^39+8x^42

The gray image is a linear code over GF(3) with n=144, k=9 and d=72.
This code was found by Heurico 1.16 in 0.18 seconds.