The generator matrix

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

generates a code of length 5 over Z3[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+60x^4+200x^6+360x^7+810x^8+740x^9+1530x^10+1620x^11+750x^12+480x^13+10x^15

The gray image is a linear code over GF(3) with n=15, k=8 and d=4.
This code was found by Heurico 1.16 in 0.016 seconds.