The generator matrix

 1  1  1  1  1  1  1  1  1  X  X  0  1  1  X  1
 0  X  0 2X^2+X  0 2X  X 2X^2+2X 2X 2X^2  0 2X^2 2X^2 X^2+X  X  0
 0  0  X X^2+2X 2X  0  X 2X^2+X 2X 2X^2+X 2X  X 2X^2+X 2X X^2+X 2X^2+X
 0  0  0 X^2  0  0 2X^2 X^2 2X^2 2X^2 2X^2  0 X^2 X^2 2X^2  0
 0  0  0  0 X^2 2X^2 X^2 X^2 X^2 2X^2 2X^2 2X^2  0 2X^2 2X^2 X^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+88x^24+102x^25+102x^26+466x^27+720x^28+294x^29+3860x^30+2400x^31+612x^32+7170x^33+2364x^34+402x^35+694x^36+234x^37+48x^38+100x^39+12x^40+14x^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 0.304 seconds.