The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X X X X X X 0 0 0
0 X 0 X 0 X 2X 2X 2X 0 0 X X 0 X 2X 2X 2X 0 0 X X 0 X 2X 2X 2X 0 0 0 X X 2X 2X X 2X X X X
0 0 X 2X 2X X 0 X 2X 0 X 2X X 2X 0 0 X 2X 0 X 2X X 2X 0 0 X 2X 0 X 2X 2X X 0 X 0 2X 0 X 2X
generates a code of length 39 over Z3[X]/(X^2) who´s minimum homogenous weight is 78.
Homogenous weight enumerator: w(x)=1x^0+72x^78+2x^81+6x^90
The gray image is a linear code over GF(3) with n=117, k=4 and d=78.
As d=78 is an upper bound for linear (117,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.00757 seconds.