The generator matrix
1 0 1 1 1 1 1 0 1 1 1 X 1 1 1 X 1 1 1 0 1 1 1 X 1 1 1 1 1 1 2X 2X 1 1 1 2X 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 X 0 X 2X 2X 2X
0 1 1 2 2X+1 0 2 1 X 2X+1 X+2 1 X X+1 X+2 1 X+1 0 2 1 X 1 X+2 1 2X 2X 2X+1 X+1 2X+2 2X+2 1 1 2X 1 2X+2 1 0 0 X 2X+1 2X+1 X+1 X X+1 0 X 2X+1 X+1 2X 2X 2X 1 1 1 2 2 X+2 X+2 2 X+2 2X+2 2X+2 2X+2 1 1 1 1 1 1 1 1 1
0 0 2X 0 X X 2X 2X 2X 0 X X X 2X 2X 2X X 2X X X 0 0 0 0 0 X 2X 0 X 2X 0 X 2X X 0 2X 0 X 2X 2X X 0 X X 2X 0 0 2X 0 X 2X 2X X 0 0 X 2X X 2X 0 0 X 2X 0 0 X 2X 2X X 0 X 2X
generates a code of length 72 over Z3[X]/(X^2) who´s minimum homogenous weight is 144.
Homogenous weight enumerator: w(x)=1x^0+234x^144+8x^162
The gray image is a linear code over GF(3) with n=216, k=5 and d=144.
As d=144 is an upper bound for linear (216,5,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.0417 seconds.