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 X X X X X X X X 0 0 0 0 1 1 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 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 2X
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 X 2X X 0 2X 0 X 2X 0 X 2X X 0 X 2X X 2X 0 0 X
generates a code of length 56 over Z3[X]/(X^2) who´s minimum homogenous weight is 112.
Homogenous weight enumerator: w(x)=1x^0+54x^112+16x^114+8x^117+2x^132
The gray image is a linear code over GF(3) with n=168, k=4 and d=112.
As d=112 is an upper bound for linear (168,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.0221 seconds.