The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  X  1  1  X  1  1  X  X  X  1  1 X^2  0  X  X  1  X X^2  0  X X^2  X  1 X^2 X^2  0  X  X X^2  X  X  X
 0 X^2  0 X^2  0 X^2  0 X^2  0 X^2  0 X^2  0 X^2  0 X^2  0 X^2 X^2  0 X^2 X^2  0 X^2 X^2  0 X^2  0 X^2  0  0 X^2 X^2 X^2  0 X^2  0  0 X^2 X^2 X^2  0  0 X^2 X^2  0 X^2  0 X^2  0  0 X^2  0

generates a code of length 53 over Z2[X]/(X^3) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+13x^56+2x^60

The gray image is a linear code over GF(2) with n=212, k=4 and d=112.
As d=112 is an upper bound for linear (212,4,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 4.
This code was found by Heurico 1.16 in 0.0513 seconds.