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- Title
Exceptions and characterization results for type‐1 λ‐designs.
- Authors
Yadav, Ajeet Kumar; Pawale, Rajendra M.; Shrikhande, Mohan S.
- Abstract
Let X be a finite set with v elements, called points and β be a family of subsets of X, called blocks. A pair (X,β) is called λ‐design whenever ∣β∣=∣X∣ and 1.for all Bi,Bj∈β,i≠j,∣Bi∩Bj∣=λ;2.for all Bj∈β,∣Bj∣=kj>λ, and not all kj are equal. The only known examples of λ‐designs are so‐called type‐1 designs, which are obtained from symmetric designs by a certain complementation procedure. Ryser and Woodall had independently conjectured that all λ‐designs are type‐1. Let r,r*(r>r*) be replication numbers of a λ‐design D=(X,β) and g=gcd(r−1,r*−1),m=gcd((r−r*)∕g,λ), and m′=m, if m is odd and m′=m∕2, otherwise. For distinct points x and y of D, let λ(x,y) denote the number of blocks of X containing x and y. We strengthen a lemma of S.S. Shrikhande and N.M. Singhi and use it to prove that if r(r−1)(v−1)−k(r−r*)m′(v−1) are not integers for k=1,2,...,m′−1, then D is type‐1. As an application of these results, we show that for fixed positive integer θ there are finitely many nontype‐1 λ‐designs with r=r*+θ. If r−r*=27 or r−r*=4p and r*≠(p−1)2, or v=7p+1 such that p≢1,13(mod21) and p≢4,9,19,24(mod35), where p is a positive prime, then D is type‐1. We further obtain several inequalities involving λ(x,y), where equality holds if and only if D is type‐1.
- Subjects
INTEGERS; EQUALITY
- Publication
Journal of Combinatorial Designs, 2020, Vol 28, Issue 9, p670
- ISSN
1063-8539
- Publication type
Article
- DOI
10.1002/jcd.21723