The generator matrix

 1  0  0  0  1  1  1  1
 0  1  0  0 X+1  X  0  1
 0  0  1  0  1  0 X+1  X
 0  0  0  1  1 X+1  X X+1

generates a code of length 8 over Z2[X]/(X^2) who´s minimum homogenous weight is 5.

Homogenous weight enumerator: w(x)=1x^0+24x^5+44x^6+40x^7+45x^8+40x^9+28x^10+24x^11+10x^12

The gray image is a linear code over GF(2) with n=16, k=8 and d=5.
As d=5 is an upper bound for linear (16,8,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 8.
This code was found by Heurico 1.16 in 0.00314 seconds.