The generator matrix

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

generates a code of length 30 over Z3[X]/(X^3) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+60x^56+24x^57+48x^59+514x^60+20x^63+48x^65+6x^66+6x^68+2x^78

The gray image is a linear code over GF(3) with n=270, k=6 and d=168.
This code was found by Heurico 1.16 in 0.401 seconds.