The generator matrix

 1  0  1  1  1  1  1  X  1  1  1 2X  1  1  1  0  1  1  1  X  1  1  1 2X  1  1  1  0  1  1  1  X  1  1  1 2X  1  1  1  1  1  1  1  1  1  0  X 2X  X  X  X  0  1  1  1  1
 0  1 2X+1  2  X X+1 X+2  1 2X  1 2X+2  1  0 2X+1  2  1  X X+1 X+2  1 2X  1 2X+2  1  0 2X+1  2  1  X X+1 X+2  1 2X  1 2X+2  1  0  X 2X 2X+1 X+1  1  2 X+2 2X+2  1  1  1  0  X 2X  X  0  X 2X+1 X+1

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+24x^112+36x^113+4x^114+12x^115+4x^120

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 0.015 seconds.