Some results on λ-designs with two block sizes

✍ Scribed by Tariq Alraqad;; Mohan Shrikhande

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 154 KB
• Volume: 19
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20272

✦ Synopsis

A k-design is a family B ={B 1 , B 2 ,...,B v } of subsets of X ={1, 2,...,v} such that |B i ∩B j |=k for all i = j and not all B i are of the same size. The only known example of k-designs (called type-1 designs) are those obtained from symmetric designs by a certain complementation procedure. Ryser [J Algebra 10 (1968), 246-261] and Woodall [Proc London Math Soc 20 (1970), 669-687] independently conjectured that all k-designs are type-1. Let g = gcd(r -1, r * -1), where r and r * are the two replication numbers. Ionin and Shrikhande [J Combin Comput 22 (1996), 135-142; J Combin Theory Ser A 74 (1996), 100-114] showed that k-designs with g = 1, 2, 3, 4 are type-1 and that the Ryser-Woodall conjecture is true for k-designs on p +1, 2p+1, 3p+1, 4p+1 points, where p is a prime. Hein and Ionin [Codes and Designs-Proceedings of Conference honoring Prof. D. K. Ray-Chaudhuri on the occasion of his 65th birthday,