The generator matrix

1 0 0 0 0 0 1 1
0 1 0 0 0 0 1 0
0 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

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+66x^2+24x^3+496x^4+440x^5+990x^6+1584x^7+990x^8+1584x^9+990x^10+440x^11+496x^12+24x^13+66x^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 1.10 in 0 seconds.