The generator matrix

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

generates a code of length 4 over Z3[X]/(X^3) 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+14706x^8+1760x^9+864x^10+192x^11+16x^12

The gray image is a linear code over GF(3) with n=36, k=9 and d=6.
This code was found by Heurico 1.16 in 0.00997 seconds.