The generator matrix
1 0 1 X
0 1 X 1
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 3.7e-005 seconds.