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Latin American Symposium on Theoretical Informatics

LATIN 2000: LATIN 2000: Theoretical Informatics pp 105–114Cite as

Rank Inequalities for Packing Designs and Sparse Triple Systems

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1776))

Abstract

Combinatorial designs find numerous applications in computer science, and are closely related to problems in coding theory. Packing designs correspond to codes with constant weight; 4-sparse partial Steiner triple systems (4-sparse PSTSs) correspond to erasure-resilient codes able to correct all (except for “bad ones”) 4-erasures, which are useful in handling failures in large disk arrays [4,10]. The study of polytopes associated with combinatorial problems has proven to be important for both algorithms and theory. However, research on polytopes for problems in combinatorial design and coding theories have been pursued only recently [14,15,17,20,21]. In this article, polytopes associated with t-(v,k,λ) packing designs and sparse PSTSs are studied. The subpacking and sparseness inequalities are introduced. These can be regarded as rank inequalities for the independence systems associated with these designs. Conditions under which subpacking inequalities define facets are studied. Sparseness inequalities are proven to induce facets for the sparse PSTS polytope; some extremal families of PSTS known as Erdös configurations play a central role in this proof. The incorporation of these inequalities in polyhedral algorithms and their use for deriving upper bounds on the packing numbers are suggested. A sample of 4-sparse PSTS (v), v ≤ 16, obtained by such an algorithm is shown; an upper bound on the size of m-sparse PSTSs is presented.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Toronto, and The Fields Institute for Research in Mathematical Sciences, Switzerland

    Lucia Moura

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  1. Lucia Moura
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Editors and Affiliations

  1. ETH Zurich, Institute of Computational Science, CH-8092 Zurich and, Swiss Institute of Bioinformatics, Switzerland

    Gaston H. Gonnet

  2. Pedeciba Informática, Casilla de Correo 16120, Distrito 6, Montevideo, Uruguay

    Alfredo Viola

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Moura, L. (2000). Rank Inequalities for Packing Designs and Sparse Triple Systems. In: Gonnet, G.H., Viola, A. (eds) LATIN 2000: Theoretical Informatics. LATIN 2000. Lecture Notes in Computer Science, vol 1776. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10719839_11

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  • DOI: https://doi.org/10.1007/10719839_11

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67306-4

  • Online ISBN: 978-3-540-46415-0

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