Designs, Codes and Cryptography

, Volume 63, Issue 3, pp 425–445 | Cite as

Divisibility of polynomials over finite fields and combinatorial applications

  • Daniel Panario
  • Olga Sosnovski
  • Brett Stevens
  • Qiang Wang


Consider a maximum-length shift-register sequence generated by a primitive polynomial f over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by f. Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polynomials. Munemasa (Finite Fields Appl 4(3):252–260, 1998) uses trinomials over \({\mathbb{F}_2}\) to construct orthogonal arrays of guaranteed strength 2 (and almost strength 3). That result was extended by Dewar et al. (Des Codes Cryptogr 45:1–17, 2007) to construct orthogonal arrays of guaranteed strength 3 by considering divisibility of trinomials by pentanomials over \({\mathbb{F}_2}\) . Here we first simplify the requirement in Munemasa’s approach that the characteristic polynomial of the sequence must be primitive: we show that the method applies even to the much broader class of polynomials with no repeated roots. Then we give characterizations of divisibility for binomials and trinomials over \({\mathbb{F}_3}\) . Some of our results apply to any finite field \({\mathbb{F}_q}\) with q elements.


Polynomials over finite fields Divisibility of polynomials Orthogonal arrays 

Mathematics Subject Classification (2000)

12E20 94A55 05B15 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Daniel Panario
    • 1
  • Olga Sosnovski
    • 1
  • Brett Stevens
    • 1
  • Qiang Wang
    • 1
  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaUK

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