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 0 0 0 0 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 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 0 X 2X X 2X
generates a code of length 42 over Z3[X]/(X^2) who´s minimum homogenous weight is 84.
Homogenous weight enumerator: w(x)=1x^0+60x^84+12x^87+8x^90
The gray image is a linear code over GF(3) with n=126, k=4 and d=84.
As d=84 is an upper bound for linear (126,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.00844 seconds.