The generator matrix

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

generates a code of length 6 over Z3[X]/(X^3) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+480x^6+54x^7+6318x^8+2758x^9+67068x^10+25272x^11+266418x^12+133974x^13+25272x^14+3714x^15+108x^16+4x^18

The gray image is a linear code over GF(3) with n=54, k=12 and d=18.
This code was found by Heurico 1.16 in 7.9 seconds.