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Packing Arrays

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LATIN 2002: Theoretical Informatics (LATIN 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2286))

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Abstract

A packing array is a b×k array of values from a g-ary alphabet such that given any two columns, i and j, and for all ordered pairs of elements from the g-ary alphabet, (g1, g2), there is at most one row, r, such that ar,i = g1 and ar,j = g2. Further, there is a set of at least n rows that pairwise differ in each column: they are disjoint. A central question is to determine, for given g and k, the maximum possible b. We developg eneral direct and recursive constructions and upper bounds on the sizes of packing arrays. We also show the equivalence of the problem to a matching problem on graphs and a class of resolvable pairwise balanced designs. We provide tables of the best known upper and lower bounds.

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References

  1. K. A. S. Abdel-Ghaffar. On the number of mutually orthogonal partial latin squares. Ars Combin., 42:259–286, 1996.

    MathSciNet  MATH  Google Scholar 

  2. K. A. S. Abdel-Ghaffar and A. E. Abbadi. Optimal disk allocation for partial match queries. ACM Transactions on Database Systems, 18(1):132–156, Mar. 1993.

    Article  Google Scholar 

  3. P. J. Cameron. Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press, Cambridge, 1994.

    Book  Google Scholar 

  4. C. Colbourn and D. Kreher. Concerning di.erence methods. Des. Codes Cryptogr., 9:61–70, 1996.

    Article  MathSciNet  Google Scholar 

  5. C. Colbourn and S. Zhao. Maximum Kirkman signal sets for synchronous uni-polar multi-user communication systems. Des. Codes Cryptogr., 20:219–227, 2000.

    Article  MathSciNet  Google Scholar 

  6. C. J. Colbourn and J. H. Dinitz, editors. The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton, 1996.

    MATH  Google Scholar 

  7. P. Danziger, M. Greig, and B. Stevens. Geometrical constructions of classuniformly resolvable structures. Unpublished Manuscript.

    Google Scholar 

  8. P. Danziger and B. Stevens. On class-uniformly resolvable frames. Unpublished Manuscript.

    Google Scholar 

  9. P. Danziger and B. Stevens. On class-uniformly resolvable groupdi visible designs. Discrete Math. (Submitted).

    Google Scholar 

  10. P. Danziger and B. Stevens. Class-uniformly resolvable designs. J. Combin. Des., 9:79–99, 2001.

    Article  MathSciNet  Google Scholar 

  11. E. Lamken, R. Rees, and S. Vanstone. Class-uniformly resolvable pairwise balanced designs with block sizes two and three. Discrete Math., 92:197–209, 1991.

    Article  MathSciNet  Google Scholar 

  12. E. Lucas. Récréations Mathématiques, volume 2. Gauthier-Villars, Paris, 1883.

    MATH  Google Scholar 

  13. N. J. A. Sloane. Covering arrays and intersecting codes. J. Combin. Des., 1:51–63, 1993.

    Article  MathSciNet  Google Scholar 

  14. B. Stevens. Transversal Covers and Packings. Ph.D. thesis, University of Toronto, Toronto, 1998.

    Google Scholar 

  15. B. Stevens, A. Ling, and E. Mendelsohn. A direct construction of transversal covers using groupdi visible designs. Ars Combin. (To Appear).

    Google Scholar 

  16. B. Stevens and E. Mendelsohn. Packing arrays and packing designs. Des. Codes Cryptogr. (To Appear).

    Google Scholar 

  17. B. Stevens and E. Mendelsohn. New recursive methods for transversal covers. J. Combin. Des., 7(3):185–203, 1999.

    Article  MathSciNet  Google Scholar 

  18. B. Stevens, L. Moura, and E. Mendelsohn. Lower bounds for transversal covers. Des. Codes Cryptogr., 15(3):279–299, 1998.

    Article  MathSciNet  Google Scholar 

  19. W. Wallis, editor. Computational and Constructive Design Theory, volume 368 of Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht, 1996.

    Google Scholar 

  20. D. Wevrick and S. A. Vanstone. Class-uniformly resolvable designs with block sizes 2 and 3. J. Combin. Des., 4:177–202, 1996.

    Article  MathSciNet  Google Scholar 

  21. Richard M. Wilson. Concerning the number of mutually orthogonal Latin squares. Discrete Math., 9:181–198, 1974.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Dr., K1S 5B6, Ottawa, ON, Canada

    Brett Stevens

  2. Department of Mathematics, University of Toronto, 100 St. George St., M6G 3G3, Toronto, ON, Canada

    Eric Mendelsohn

Authors
  1. Brett Stevens
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  2. Eric Mendelsohn
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Editor information

Editors and Affiliations

  1. Compaq Cambridge Research Laboratory, One Cambridge Center, 02142-1612, Cambridge, MA, USA

    Sergio Rajsbaum

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© 2002 Springer-Verlag Berlin Heidelberg

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Stevens, B., Mendelsohn, E. (2002). Packing Arrays. In: Rajsbaum, S. (eds) LATIN 2002: Theoretical Informatics. LATIN 2002. Lecture Notes in Computer Science, vol 2286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45995-2_28

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  • DOI: https://doi.org/10.1007/3-540-45995-2_28

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43400-9

  • Online ISBN: 978-3-540-45995-8

  • eBook Packages: Springer Book Archive

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