The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+48x^2+428x^4+1936x^6+3366x^8+1936x^10+428x^12+48x^14+1x^16

The gray image is a linear code over GF(2) with n=16, k=13 and d=2.
As d=2 is an upper bound for linear (16,13,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 13.
This code was found by Heurico 1.16 in 0.0546 seconds.