Abstract
Strong external difference families (SEDFs) and their generalizations GSEDFs and BGSEDFs in a finite abelian group G are combinatorial designs introduced by Paterson and Stinson (Discret Math 339: 2891–2906, 2016) and have applications in communication theory to construct optimal strong algebraic manipulation detection codes. In this paper we firstly present some general constructions of these combinatorial designs by using difference sets and partial difference sets in G. Then, as applications of the general constructions, we construct series of SEDF, GSEDF and BGSEDF in finite fields by using cyclotomic classes. Particularly, we present an \((n,m,k,\lambda )=(243,11,22,20)\)-SEDF in \((\mathbb {F}_q,+)\ (q=3^5=243)\) by using the cyclotomic classes of order 11 in \(\mathbb {F}_q\) which answers an open problem raised in Paterson and Stinson (2016).
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Acknowledgements
The authors are grateful to the two anonoymous reviewers for their detailed comments and suggestions that much improved the presentation and quality of this paper. The work of J. Wen and F. Fu was supported by the National Key Basic Research Program of China under Grant 2013CB834204, and the NSFC under Grant 61571243, 61171082. The work of M. Yang was supported by the NSFC under Grant 61379139, 11526215, 11501156, and the Natural Science Research Project of Higher Education of Anhui Province of China under Grant KJ 2015JD18. The work of K. Feng was supported by the NSFC under Grant 11571007 and 11471178.
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Communicated by M. Paterson.
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Wen, J., Yang, M., Fu, F. et al. Cyclotomic construction of strong external difference families in finite fields. Des. Codes Cryptogr. 86, 1149–1159 (2018). https://doi.org/10.1007/s10623-017-0384-y
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DOI: https://doi.org/10.1007/s10623-017-0384-y
Keywords
- Strong external difference family
- Difference set
- Partial difference set
- Cyclotomic class
- Cyclotomic number
- Finite field
- Strong algebraic manipulation detection code