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Designs, Codes and Cryptography

, Volume 86, Issue 5, pp 1095–1111 | Cite as

Constructing permutation arrays from groups

  • Sergey Bereg
  • Avi Levy
  • I. Hal Sudborough
Article

Abstract

Let M(nd) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M(nd). We compute the Hamming distances of affine semilinear groups and projective semilinear groups, and unions of cosets of AGL(1, q) and PGL(2, q) with Frobenius maps to obtain new, improved lower bounds for M(nd). We give new randomized algorithms. We give better lower bounds for M(nd) also using new theorems concerning the contraction operation. For example, we prove a quadratic lower bound for \(M(n,n-2)\) for all \(n\equiv 2 \pmod 3\) such that \(n+1\) is a prime power.

Keywords

Permutation codes Permutation arrays Finite fields Groups 

Mathematics Subject Classification

05A05 94B25 05E18 

References

  1. 1.
    Bereg S., Morales L., Sudborough I.H.: Parallel and sequential partition and extension techniques. Manuscript, The University of Texas Dallas (2016).Google Scholar
  2. 2.
    Bereg S., Morales L., Sudborough I.H.: Extending permutation arrays: improving MOLS bounds. Des. Codes Cryptogr. 83(3), 661–683 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blake I.F., Cohen G.D., Deza M.: Coding with permutations. Inform. Control 43(1), 1–19 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cameron P.: Permutation Groups. London Mathematical Society Student TextsCambridge University Press, Cambridge (1999).CrossRefzbMATHGoogle Scholar
  5. 5.
    Chu W., Colbourn C.J., Dukes P.: Constructions for permutation codes in powerline communications. Des. Codes Cryptogr. 32(1–3), 51–64 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Colbourn C.J., Kløve T., Ling A.C.H.: Permutation arrays for powerline communication and mutually orthogonal latin squares. IEEE Trans. Inform. Theory 50(6), 1289–1291 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A.: Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon Press, Oxford (1985).zbMATHGoogle Scholar
  8. 8.
    Dixon J., Mortimer B.: Permutation Groups. Springer, New York (1996).CrossRefzbMATHGoogle Scholar
  9. 9.
    Dummit D.S., Foote R.M.: Abstract Algebra, 2nd edn. Wiley, New York (1999).zbMATHGoogle Scholar
  10. 10.
    Frankl P., Deza M.: On the maximum number of permutations with given maximal or minimal distance. J. Comb. Theory Ser. A 22(3), 352–360 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gao F., Yang Y., Ge G.: An improvement on the Gilbert-Varshamov bound for permutation codes. IEEE Trans. Inform. Theory 59(5), 3059–3063 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Herstein I.N.: Topics in Algebra, 2nd edn. Wiley, New York (1975).zbMATHGoogle Scholar
  13. 13.
    Huczynska S.: Powerline communication and the 36 officers problem. Philos. Trans. R. Soc. Lond. A 364(1849), 3199–3214 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Janiszczak I., Lempken W., Östergård P.R.J., Staszewski R.: Permutation codes invariant under isometries. Des. Codes Cryptogr. 75(3), 497–507 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Janiszczak I., Staszewski R.: An improved bound for permutation arrays of length 10. Technical Report 4, Institute for Experimental Mathematics, University Duisburg-Essen (2008).Google Scholar
  16. 16.
    Lidl R., Niederreiter H.: Finite Fields, 2nd edn. Cambridge University Press, Cambridge (1997).zbMATHGoogle Scholar
  17. 17.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland/Elsevier, Amsterdam (1977).zbMATHGoogle Scholar
  18. 18.
    Passman D.: Sharp transitivity. In: Permutation Groups. Benjamin, Inc., New York (1968).Google Scholar
  19. 19.
    Pavlidou N., Vinck A.H., Yazdani J., Honary B.: Power lines communications: state of the art and future trends. IEEE Commun. Mag. 34–40 (2003).Google Scholar
  20. 20.
    Pless V.S., H W.C. (eds.): Handbook of Coding Theory. North-Holland/Elsevier, Amsterdam (1998).zbMATHGoogle Scholar
  21. 21.
    Smith D.H., Montemanni R.: A new table of permutation codes. Des. Codes Cryptogr. 63(2), 241–253 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
  23. 23.
    Yang L., Chen K., Yuan L.: New constructions of permutation arrays. arXiv:abs/0801.3987 (2008).
  24. 24.
    Yang L., Chen K., Yuan L.: New lower bounds on sizes of permutation arrays. arXiv:abs/0801.3986 (2008).

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Computer Science, Erik Jonsson School of Engineering and Computer ScienceUniversity of Texas at DallasRichardsonUSA
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA

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