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 1 1 1 1 1 1 1 1 1 1
0 X 0 0 X 0 X X 2X 0 0 X X 0 2X X 2X 2X 0 X 2X 0 X 2X 2X 2X 2X 0 0 0 X X 0 X X 2X 0 0 X X 0 2X X 2X 2X 0 X 2X 0 X 2X 2X 2X 2X 0
0 0 X 0 2X X 2X X 2X 0 X X 0 2X 0 0 X X 2X 2X 2X 2X X 0 0 X 2X 0 0 X 2X X X 2X 0 2X 0 X X 0 2X 0 0 X X 2X 2X 2X 2X X 0 0 X 2X 0
0 0 0 X 2X 2X 0 2X 2X 2X X 0 2X 0 2X X X 0 X X X 2X X 0 X 2X 0 0 X 2X 2X 2X X X X 2X 2X 0 0 0 X X 2X 2X X 0 0 0 2X X 0 2X 0 X 0
generates a code of length 55 over Z3[X]/(X^2) who´s minimum homogenous weight is 108.
Homogenous weight enumerator: w(x)=1x^0+26x^108+162x^110+52x^111+2x^165
The gray image is a linear code over GF(3) with n=165, k=5 and d=108.
As d=108 is an upper bound for linear (165,5,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.0169 seconds.