The generator matrix
1 1 1 1 1 1
0 X 0 0 2X^2+X X^2
0 0 X 0 2X^2+2X X^2
0 0 0 X 2X^2+2X X^2+2X
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+62x^6+78x^7+270x^8+230x^9+792x^10+1080x^11+14178x^12+1422x^13+1080x^14+346x^15+138x^16+6x^18
The gray image is a linear code over GF(3) with n=54, k=9 and d=18.
This code was found by Heurico 1.16 in 0.0304 seconds.