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 1 1 1 X X X X X X 0 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 0 X 2X X X 2X 2X 0 X X 0 2X 0 X X 0
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 0 0 2X X 0 X 2X 0 0 0 2X X X 2X X
generates a code of length 77 over Z3[X]/(X^2) who´s minimum homogenous weight is 155.
Homogenous weight enumerator: w(x)=1x^0+54x^155+12x^156+12x^159+2x^162
The gray image is a linear code over GF(3) with n=231, k=4 and d=155.
As d=155 is an upper bound for linear (231,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.0545 seconds.