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Designs, Codes and Cryptography

, Volume 83, Issue 3, pp 661–683 | Cite as

Extending permutation arrays: improving MOLS bounds

  • Sergey Bereg
  • Linda Morales
  • I. Hal SudboroughEmail author
Article

Abstract

A permutation array (PA) A is a set of permutations on \(Z_n=\{0,1,\dots ,n-1\}\), for some n. A PA A has pairwise Hamming distance at least d, if for every pair of permutations \(\sigma \) and \(\tau \) in A, there are at least d integers i in \(Z_n\) such that \(\sigma (i)\ne \tau (i)\). Let M(nd) denote the maximum number of permutations in any PA with pairwise Hamming distance at least d. Recently considerable effort has been devoted to improving known lower bounds for M(nd) for all \(n>d>3\). We give a partition and extension operation that enables the production of a new PA \(A'\) for \(M(n+1,d)\) from an existing PA A for \(M(n,d-1)\). In particular, this operation allows for improvements for PA’s for \(M(q+1,q)\) for powers of prime numbers q, as well as for many other choices of n and d, where n is not a power of a prime. Finally, for prime numbers p, the partition and extension technique provides an asymptotically better lower bound for \(M(p+1,p)\) than that given by current knowledge about mutually orthogonal Latin squares. We prove a new asymptotic lower bound for the set of primes p, namely, \(M(p+1,p)\ge p^{1.5}/2-O(p)\).

Keywords

Permutation arrays Hamming distance Error correcting codes 

Mathematics Subject Classification

05A05 05A18 68R05 94B60 

Notes

Acknowledgments

We would like to thank the referees for their many helpful suggestions and valuable comments.

References

  1. 1.
    Bereg S., Morales L., Sudborough I.: Table of lower bounds for \(M(n,n-1)\). http://www.utdallas.edu/~sxb027100/soft/permutation-arrays/n-1/ (2016).
  2. 2.
    Bereg S., Morales L., Sudborough I.: PAs exhibiting results from partition and extension. http://www.utdallas.edu/~sxb027100/soft/permutation-arrays/n-1/data.zip (2016).
  3. 3.
    Beth T.: Eine Bemerkung zur Abschätzung der Anzahl orthogonaler lateinischer Quadrate mittels Siebverfahren. Abh. Math. Sem. Hamburg. 53, 284–288 (1983).Google Scholar
  4. 4.
    Chu W., Colbourn C.J., Dukes P.: Constructions for permutation codes in powerline communications. Des. Codes Cryptogr. 32, 51–64 (2004).Google Scholar
  5. 5.
    Colbourn C.J., Dinitz J.H.: Handbook of Combinatorial Designs, 2nd edn. Chapman and Hall/CRC, New York (2006).Google Scholar
  6. 6.
    Colbourn C.J., Kløve T., Ling A.C.H.: Permutation arrays for powerline communication and mutually orthogonal latin squares. IEEE Trans. Inf. Theory 50, 1289–1291 (2004).Google Scholar
  7. 7.
    Deza M., Vanstone S.A.: Bounds for permutation arrays. J. Stat. Plan. Inference 2, 197–209 (1978).Google Scholar
  8. 8.
    Frankl P., Deza M.: On the maximum number of permutations with given maximal or minimal distance. J. Comb. Theory A 22, 352–360 (1977).Google Scholar
  9. 9.
    Gao F., Yang Y., Ge G.: An improvement on the Gilbert-Varshamov bound for permutation codes. IEEE Trans. Inf. Theory 59, 3059–3063 (2013).Google Scholar
  10. 10.
    Huang Y.-Y., Tsai S.-C., Wu H.-L.: On the construction of permutation arrays via mappings from binary vectors to permutations. Des. Codes Cryptogr. 40, 139–155 (2006).Google Scholar
  11. 11.
    Huczynska S.: Powerline communication and the 36 officers problem. Philos. Trans. R. Soc. Lond. A 364, 3199–3214 (2006).Google Scholar
  12. 12.
    Janiszczak I., Staszewski R.: An Improved Bound for Permutation Arrays of Length 10. Technical Report 4. Institute for Experimental Mathematics, University Duisburg-Essen, Essen (2008).Google Scholar
  13. 13.
    Janiszczak I., Lempken W., Östergård P.R.J., Staszewski R.: Permutation codes invariant under isometries. Des. Codes Cryptogr. 75, 497–507 (2015).Google Scholar
  14. 14.
    Lin T.-T., Tsai S.-C., Tzeng W.-G.: Efficient encoding and decoding with permutation arrays. In: IEEE International Symposium on Information Theory (ISIT’08), pp. 211–214 (2008).Google Scholar
  15. 15.
    Nguyen Q.T.: Transitivity and hamming distance of permutation arrays. PhD thesis, University of Texas at Dallas Richardson, TX (2013).Google Scholar
  16. 16.
    Passman D.: Permutation Groups. Benjamin Inc, New York (1968).Google Scholar
  17. 17.
    Pavlidou N., Vinck A.H., Yazdani J., Honary B.: Power line communications: state of the art and future trends. IEEE Commun. Mag. 41, 34–40 (2003).Google Scholar
  18. 18.
    Quistorff J.: A new nonexistence result for sharply multiply transitive permutation sets. Discret. Math. 288, 185–186 (2004).Google Scholar
  19. 19.
    Rotman J.J.: An Introduction to the Theory of Groups, 4th edn. Springer, New York (1995).Google Scholar
  20. 20.
    Smith D.H., Montemanni R.: A new table of permutation codes. Des. Codes Cryptogr. 63, 241–253 (2012).Google Scholar
  21. 21.
    The Sage Developers: Mutually Orthogonal Latin Squares (MOLS). http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/designs/latin_squares.html.

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sergey Bereg
    • 1
  • Linda Morales
    • 1
  • I. Hal Sudborough
    • 1
    Email author
  1. 1.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA

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