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Journal of Algebraic Combinatorics

, Volume 49, Issue 1, pp 21–48 | Cite as

Construction and nonexistence of strong external difference families

  • Jonathan JedwabEmail author
  • Shuxing Li
Article

Abstract

Strong external difference families (SEDFs) were introduced by Paterson and Stinson as a more restrictive version of external difference families. SEDFs can be used to produce optimal strong algebraic manipulation detection codes. We characterize the parameters \((v, m, k, \lambda )\) of a nontrivial SEDF that is near-complete (satisfying \(v=km+1\)). We construct the first known nontrivial example of a \((v, m, k, \lambda )\) SEDF having \(m > 2\). The parameters of this example are (243, 11, 22, 20), giving a near-complete SEDF, and its group is \(\mathbb {Z}_3^5\). We provide a comprehensive framework for the study of SEDFs using character theory and algebraic number theory, showing that the cases \(m=2\) and \(m>2\) are fundamentally different. We prove a range of nonexistence results, greatly narrowing the scope of possible parameters of SEDFs.

Keywords

Character sum Exponent bound Finite projective geometry Mathieu group Partial difference set Strong external difference family 

Notes

Acknowledgements

We are grateful to Ruizhong Wei for kindly supplying a preprint of the paper [3]. We thank the referee for providing very careful and helpful comments.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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