Hasse–Weil bound for additive cyclic codes
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We obtain a bound on the minimum distance of additive cyclic codes via the number of rational points on certain algebraic curves over finite fields. This is an extension of the analogous bound in the case of classical cyclic codes. Our result is the only general bound on such codes aside from Bierbrauer’s BCH bound. We compare our bounds’ performance against the BCH bound for additive cyclic codes in a special case and provide examples where it yields better results.
KeywordsAdditive cyclic code Algebraic curve over a finite field Hasse–Weil bound BCH bound
Mathematics Subject Classification94B27 94B65
The first author was supported by TÜBİTAK Project 114F432. The third author was supported by TÜBİTAK BİDEB 2211 National Ph.D. Scholarship Programme. The authors thank Kamil Otal for his help with Magma computations. We also thank the Reviewers for useful suggestions that improved the manuscript. In particular, Remark 8 was written in response to a question by one of the reviewers.
- 1.Bosma W., Cannon J., Playoust, C.: The magma algebra system I. The user language. J. Symb. Comput. 24, 235–265 (1997).Google Scholar
- 2.Bierbrauer J.: The theory of cyclic codes and a generalization to additive codes. Des. Codes Cryptogr. 25, 189–206 (2002).Google Scholar
- 3.Bierbrauer J.: Introduction to Coding Theory. Chapman and Hall/CRC Press, Boca Raton (2005).Google Scholar
- 4.Bierbrauer J., Edel Y.: Quantum twisted codes. J. Comb. Des. 8, 174–188 (2000).Google Scholar
- 5.Garcia A., Özbudak F.: Some maximal function fields and additive polynomials. Commun. Algebra 35, 1553–1566 (2007).Google Scholar
- 6.Güneri C., Özbudak F.: Weil-Serre type bounds for cyclic codes. IEEE Trans. Inf. Theory 54, 5381–5395 (2008).Google Scholar
- 7.Stichtenoth H.: Algebraic Function Fields and Codes. Springer GTM, New York 254, (2009).Google Scholar
- 8.Wolfmann, J.: New bounds on cyclic codes from algebraic curves. Coding Theory and Applications (Toulon, 1988). Lecture Notes in Computer Science. vol. 388, pp. 47–62 (1989).Google Scholar