The generator matrix

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

generates a code of length 4 over Z3[X]/(X^3) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+36x^4+84x^6+144x^7+1620x^8+152x^9+144x^10+6x^12

The gray image is a linear code over GF(3) with n=36, k=7 and d=12.
This code was found by Heurico 1.16 in 0.00032 seconds.