The generator matrix

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

generates a code of length 16 over Z3[X]/(X^3) who´s minimum homogenous weight is 27.

Homogenous weight enumerator: w(x)=1x^0+116x^27+162x^30+1458x^32+324x^33+114x^36+12x^45

The gray image is a linear code over GF(3) with n=144, k=7 and d=81.
This code was found by Heurico 1.16 in 0.162 seconds.