The generator matrix
1 0 0 1
0 1 0 2X^2+2X
0 0 1 2X^2+2X
0 0 0 2X^2+2X+1
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+96x^2+8x^3+3456x^4+576x^5+55320x^6+13824x^7+332928x^8+110624x^9+13824x^10+768x^11+16x^12
The gray image is a linear code over GF(3) with n=36, k=12 and d=6.
This code was found by Heurico 1.16 in 2.58 seconds.