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New lower bounds for permutation arrays using contraction

  • Sergey Bereg
  • Zevi MillerEmail author
  • Luis Gerardo Mojica
  • Linda Morales
  • I. H. Sudborough
Article
  • 22 Downloads

Abstract

A permutation array A is a set of permutations on a finite set \(\Omega \), say of size n. Given distinct permutations \(\pi , \sigma \in \Omega \), we let \(hd(\pi , \sigma ) = |\{ x\in \Omega : \pi (x) \ne \sigma (x) \}|\), called the Hamming distance between \(\pi \) and \(\sigma \). Now let \(hd(A) =\) min\(\{ hd(\pi , \sigma ): \pi , \sigma \in A \}\). For positive integers n and d with \(d\le n\) we let M(nd) be the maximum number of permutations in any array A satisfying \(hd(A) \ge d\). There is an extensive literature on the function M(nd), motivated in part by suggested applications to error correcting codes for message transmission over power lines. A basic fact is that if a permutation group G is sharply k-transitive on a set of size \(n\ge k\), then \(M(n,n-k+1) = |G|\). Motivated by this we consider the permutation groups AGL(1, q) and PGL(2, q) acting sharply 2-transitively on GF(q) and sharply 3-transitively on \(GF(q)\cup \{\infty \}\) respectively. Applying a contraction operation to these groups, we obtain the following new lower bounds for prime powers q satisfying \(q\equiv 1\) (mod 3).
  1. 1.

    \(M(q-1,q-3)\ge (q^{2} - 1)/2\) for q odd, \(q\ge 7\),

     
  2. 2.

    \(M(q-1,q-3)\ge (q-1)(q+2)/3\) for q even, \(q\ge 8\),

     
  3. 3.

    \(M(q,q-3)\ge Kq^{2}log(q)\) for some constant \(K>0\) if q is odd.

     
These results resolve a case left open in a previous paper (Bereg et al. in Des Codes Cryptogr 86(5):1095–1111, 2018), where it was shown that \(M(q-1, q-3) \ge q^{2} - q\) and \(M(q,q-3) \ge q^{3} - q\) for all prime powers q such that \(q\not \equiv 1\) (mod 3). We also obtain lower bounds for M(nd) for a finite number of exceptional pairs nd, by applying this contraction operation to the sharply 4 and 5-transitive Mathieu groups.

Keywords

Permutation arrays Contraction AGL(1, q) PGL(2, q) 

Mathematics Subject Classification

05A05 05B30 05C25 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Sergey Bereg
    • 1
  • Zevi Miller
    • 2
    Email author
  • Luis Gerardo Mojica
    • 1
  • Linda Morales
    • 1
  • I. H. Sudborough
    • 1
  1. 1.Computer Science DepartmentUniversity of Texas at DallasRichardsonUSA
  2. 2.Department of MathematicsMiami UniversityOxfordUSA

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