The generator matrix

 1  1  1
 0  X  0
 0  0  X

generates a code of length 3 over Z3[X]/(X^2) who´s minimum homogenous weight is 3.

Homogenous weight enumerator: w(x)=1x^0+6x^3+66x^6+8x^9

The gray image is a linear code over GF(3) with n=9, k=4 and d=3.
This code was found by Heurico 1.16 in 4.2e-005 seconds.