The generator matrix

 1  1  1  1
 0  X  0  X
 0  0  X  X

generates a code of length 4 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+14x^4+1x^8

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^2) for dimension 4.
This code was found by Heurico 1.16 in 4.89e-005 seconds.