The generator matrix

 1  0  1  1
 0  1 2X+1  X

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+14x^6+12x^7+18x^8+28x^9+6x^10+2x^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 4.01e-005 seconds.