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 1 1 1 1 1 1 1 1 1 X X X X X X X X 0 0 0 0 1 1 1 1 X 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 X X 0 X 2X 2X 2X 0 0 X X 0 X 2X 2X 2X 0 X X X 0 2X 2X 2X X X X 0 0 0 X X X 0 X 2X X 2X 0 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 X 2X 0 0 X 2X 0 X 2X X 2X 0 0 X 2X X 2X X 0 2X 0 X 2X 0 X 2X X 0 X 2X X 2X 2X 0 X X 2X 0 0 0 0 X 2X
generates a code of length 73 over Z3[X]/(X^2) who´s minimum homogenous weight is 147.
Homogenous weight enumerator: w(x)=1x^0+72x^147+6x^153+2x^162
The gray image is a linear code over GF(3) with n=219, k=4 and d=147.
As d=147 is an upper bound for linear (219,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.0602 seconds.