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


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, \(M(p+1,p)\ge p^{1.5}/2-O(p)\).


Permutation arrays Hamming distance Error correcting codes 

Mathematics Subject Classification

05A05 05A18 68R05 94B60 



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


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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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