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

generates a code of length 60 over Z3[X]/(X^2) who´s minimum homogenous weight is 120.

Homogenous weight enumerator: w(x)=1x^0+72x^120+2x^126+4x^135+2x^144

The gray image is a linear code over GF(3) with n=180, k=4 and d=120.
As d=120 is an upper bound for linear (180,4,3)-codes, this code is optimal over Z3[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.0266 seconds.