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.