The generator matrix

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

generates a code of length 14 over Z2[X]/(X^2) who´s minimum homogenous weight is 8.

Homogenous weight enumerator: w(x)=1x^0+386x^8+500x^10+2132x^12+2236x^14+1873x^16+812x^18+212x^20+36x^22+4x^24

The gray image is a linear code over GF(2) with n=28, k=13 and d=8.
As d=8 is an upper bound for linear (28,13,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 13.
This code was found by Heurico 1.10 in 0.141 seconds.