The generator matrix

 1  1  1  1  1  1  1  X  1  1  X  0  X  1  1  1
 0  X 2X  0 2X^2+X 2X  0 2X^2+X 2X^2+X 2X 2X  X 2X^2+X X^2  0  0
 0  0 X^2  0  0  0  0 2X^2 X^2 X^2 2X^2 X^2 2X^2  0  0 X^2
 0  0  0 X^2  0  0 X^2 2X^2 2X^2 2X^2  0 X^2 X^2  0  0 X^2
 0  0  0  0 2X^2  0 2X^2 2X^2 2X^2  0 2X^2 2X^2 X^2 2X^2  0  0
 0  0  0  0  0 X^2 2X^2  0 2X^2 X^2 X^2 X^2 X^2 2X^2 X^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+24x^23+76x^24+240x^26+336x^27+486x^28+960x^29+3340x^30+1944x^31+1758x^32+6616x^33+1944x^34+1272x^35+430x^36+120x^38+88x^39+4x^42+16x^45+4x^48

The gray image is a linear code over GF(3) with n=144, k=9 and d=63.
This code was found by Heurico 1.16 in 0.271 seconds.