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 1 1 1 1 1 1 X X X X X X X X 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 X 1 X 0 X X 0 1 1 1
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 X X 0 X 2X 2X 2X 0 X X X 0 2X 2X 2X 0 X X X 0 0 X X 0 X 2X 2X 2X 0 0 X X 2X X 0 0 0 2X X 2X
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 X 2X 0 0 X 2X X 2X X 0 2X 0 X 2X 0 0 X 2X X X 2X X 2X 0 0 X 2X 0 X 2X X X 2X X 2X X 0 2X 2X
generates a code of length 69 over Z3[X]/(X^2) who´s minimum homogenous weight is 138.
Homogenous weight enumerator: w(x)=1x^0+66x^138+4x^144+6x^147+4x^153
The gray image is a linear code over GF(3) with n=207, k=4 and d=138.
As d=138 is an upper bound for linear (207,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.0495 seconds.