The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+4x^2+22x^4+4x^6+1x^8

The gray image is a linear code over GF(2) with n=8, k=5 and d=2.
As d=2 is an upper bound for linear (8,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 2.19e-005 seconds.