The generator matrix

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

generates a code of length 4 over Z3[X]/(X^2) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+12x^6+54x^8+8x^9+6x^12

The gray image is a linear code over GF(3) with n=12, k=4 and d=6.
As d=6 is an upper bound for linear (12,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 5.89e-005 seconds.