The generator matrix
1 1 1 1 1 1 1
0 X 0 X 0 X 2X
0 0 X 2X 2X X 0
generates a code of length 7 over Z3[X]/(X^2) who´s minimum homogenous weight is 12.
Homogenous weight enumerator: w(x)=1x^0+10x^12+54x^14+12x^15+2x^18+2x^21
The gray image is a linear code over GF(3) with n=21, k=4 and d=12.
As d=12 is an upper bound for linear (21,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.000107 seconds.