The generator matrix

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

generates a code of length 4 over Z3[X]/(X^2) who´s minimum homogenous weight is 3.

Homogenous weight enumerator: w(x)=1x^0+8x^3+108x^4+240x^6+432x^7+486x^8+464x^9+432x^10+16x^12

The gray image is a linear code over GF(3) with n=12, k=7 and d=3.
This code was found by Heurico 1.13 in 6.6e-005 seconds.