The generator matrix

 1  0  1
 0  1  X

generates a code of length 3 over Z2[X]/(X^2) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+3x^2+8x^3+3x^4+1x^6

The gray image is a linear code over GF(2) with n=6, k=4 and d=2.
As d=2 is an upper bound for linear (6,4,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 1.79e-005 seconds.