The lotto numbers L(n,3,p,2)

✍ Scribed by Nicolas Bougard

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 155 KB
• Volume: 14
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20075

✦ Synopsis

Abstract

An (n,k,p,t)‐lotto design is an n‐set N and a set ${\cal B}$ of k‐subsets of N (called blocks) such that for each p‐subset P of N, there is a block $B \in {\cal B}$ for which $\left | P \cap B \right | \geq t$. The lotto number L(n,k,p,t) is the smallest number of blocks in an (n,k,p,t)‐lotto design. The numbers C(n,k,t) = L(n,k,t,t) are called covering numbers. It is easy to show that, for n ≥ k(p − 1),

For k = 3, we prove that equality holds if one of the following holds:

n is large, in particular $ n\geq \big { {\matrix{(p - 1)(2p - 3)\quad {\rm if }n \not \equiv p(\bmod ;2),\cr (p - 1)(4p - 8)\quad {\rm if }n \equiv p(\bmod ;2),}}$

$n \equiv p - 4,p - 3, \ldots ,3p - 1(\bmod 6(p - 1)),$

2 ≤ p ≤ 6.

© 2006 Wiley Periodicals, Inc. J Combin Designs 14: 333–350, 2006