combinatorics problem

My problem is a little difficulty, i explain it with en example.

I have the following matrix
1. 1 1 1 1 0 1 1 0 1 0 0 0 0 0 1 1 0
2. 1 0 1 1 1 1 1 0 1 0 0 0 0 1 1 0 0
3. 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1
4. 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 1
5. 0 1 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1
6. 1 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 1
7. 1 0 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1
8. 1 0 1 1 0 0 0 0 0 0 1 1 1 0 1 1 1
9. 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 1
10. 1 0 0 1 0 0 0 1 1 0 1 0 1 1 1 1 1

And i am searching a vector of combinations in the rows that gave me
a non zero line. for example the combination with 1 4 6 is in the sum
a non zero vector

1. 1 1 1 1 0 1 1 0 1 0 0 0 0 0 1 1 0
4. 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 1
= 2 1 1 2 1 1 2 1 1 1 1 0 0 0 1 2 1
6. 1 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 1
= 3 2 1 2 1 2 3 2 2 1 1 1 1 1 2 2 2

Now i have different accretions. One is the brute force that takes a
lot of time and a other the simplex algo. But right now i have made
the brute force thing and it takes a lot of time. Maybe some one of
you have a better idea to make that thing work. Thank you for every
idea or help!! =)
.

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