# New Bounds for Covering Arrays of Strength 7

Happy Halloween!

I want to share some cool results of a project I’m working on in one of my classes. A covering array is a 4-tuple ${CA(N; t, k, v)}$ which is an ${N \times k}$ array, each entry is from an alphabet of size ${v}$, and for every ${t}$ of the ${k}$ columns, all ${t}$-tuples over ${v}$ exists in at least one row when restricted to these columns. The covering array number, ${CAN(t, k, v)}$, is the smallest ${N}$ for which a ${CA(N; t, k, v)}$ exists. Kleitman and Spencer, and Katona independently, found ${CAN(2, k, 2)}$ for all ${k}$; no other cases are known for all ${k}$, and only heuristics are known. My advisor keeps the best-known covering array numbers here.

I was able to show the following:

1. ${CAN(7, 10, 3) \le 4371 (-2184)}$,
2. ${CAN(7, 11, 3) \le 6555 (-2184)}$,
3. ${CAN(7, 13, 3) \le 9225 (-1698)}$,
4. ${CAN(7, 14, 3) \le 10923 (-2184)}$,
5. ${CAN(7, 15, 3) \le 13107 (-2184)}$,
6. ${CAN(7, 17, 5) \le 312485 (-78120)}$.

The numbers in parentheses are the row reductions from previous known bounds. I won’t share how I did this yet, but it is a cool computational technique that works very well for high ${t}$!