The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+24x^2+8x^3+216x^4+144x^5+888x^6+864x^7+1584x^8+1760x^9+864x^10+192x^11+16x^12

The gray image is a linear code over GF(3) with n=12, k=8 and d=2.
This code was found by Heurico 1.16 in 0.00189 seconds.