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 X X X X X X X X 1 1 1 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 X X 0 X 2X 2X 2X 0 0 X X 0 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 X 2X X 2X 0 0 X 2X 0 X 2X X 2X 2X
generates a code of length 41 over Z3[X]/(X^2) who´s minimum homogenous weight is 81.
Homogenous weight enumerator: w(x)=1x^0+58x^81+6x^84+6x^87+8x^90+2x^99
The gray image is a linear code over GF(3) with n=123, k=4 and d=81.
As d=81 is an upper bound for linear (123,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.00533 seconds.