Constructions and uses of incomplete pairwise balanced designs

We give explicit constructions for incomplete pairwise balanced designs IPBD(( v ;  w ),  K ), or, equivalently, edge-decompositions of a difference of two cliques K v \ K w into cliques whose sizes belong to the set K . Our constructions produce such designs whenever v and w satisfy the usual divis...

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Bibliographic Details
Published in:Designs, codes, and cryptography codes, and cryptography, 2019-12, Vol.87 (12), p.2729-2751
Main Authors: Dukes, Peter J., Lamken, Esther R.
Format: Article
Language:English
Subjects:
Arrays
Circuits
Coding and Information Theory
Computer Science
Cryptology
Data Structures and Information Theory
Discrete Mathematics in Computer Science
Information and Communication
Latin square design
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Summary:We give explicit constructions for incomplete pairwise balanced designs IPBD(( v ;  w ),  K ), or, equivalently, edge-decompositions of a difference of two cliques K v \ K w into cliques whose sizes belong to the set K . Our constructions produce such designs whenever v and w satisfy the usual divisibility conditions, have ratio v  /  w bounded away from the smallest value in K minus one, say v / w > k - 1 + ϵ , for k = min K and ϵ > 0 , and are sufficiently large (depending on K and ϵ ). As a consequence, some new results are obtained on many related designs, including class-uniformly resolvable designs, incomplete mutually orthogonal latin squares, and group divisible designs. We also include several other applications that illustrate the power of using IPBDs as ‘templates’.
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-019-00645-6