The generator matrix

 1  1
 0 X^2

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

Homogenous weight enumerator: w(x)=1x^0+14x^2+1x^4

The gray image is a linear code over GF(2) with n=8, k=4 and d=4.
As d=4 is an upper bound for linear (8,4,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 4.
This code was found by Heurico 1.16 in 3.5e-005 seconds.