The generator matrix

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

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+84x^24+18x^26+146x^27+54x^28+108x^29+1278x^30+216x^31+1674x^32+2292x^33+216x^34+144x^35+222x^36+90x^39+18x^42

The gray image is a linear code over GF(3) with n=144, k=8 and d=72.
This code was found by Heurico 1.16 in 0.0399 seconds.