The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 X X 1 1 X 1 X X X
0 X^2 2X^2 0 X^2 2X^2 0 X^2 2X^2 0 X^2 2X^2 X^2 2X^2 0 X^2 0 2X^2 X^2 2X^2 0
generates a code of length 21 over Z3[X]/(X^3) who´s minimum homogenous weight is 42.
Homogenous weight enumerator: w(x)=1x^0+72x^42+2x^45+6x^48
The gray image is a linear code over GF(3) with n=189, k=4 and d=126.
As d=126 is an upper bound for linear (189,4,3)-codes, this code is optimal over Z3[X]/(X^3) for dimension 4.
This code was found by Heurico 1.16 in 0.00143 seconds.