The generator matrix
1 1 1 1
0 X 0 0
0 0 X 0
0 0 0 X
generates a code of length 4 over Z3[X]/(X^2) who´s minimum homogenous weight is 3.
Homogenous weight enumerator: w(x)=1x^0+8x^3+24x^6+162x^8+32x^9+16x^12
The gray image is a linear code over GF(3) with n=12, k=5 and d=3.
This code was found by Heurico 1.16 in 7.61e-005 seconds.