The generator matrix

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

generates a code of length 4 over Z2[X]/(X^3) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+13x^4+2x^6

The gray image is a linear code over GF(2) with n=16, k=4 and d=8.
As d=8 is an upper bound for linear (16,4,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 4.
This code was found by Heurico 1.16 in 6.1e-005 seconds.