Geometric Approach to b-Symbol Hamming Weights of Cyclic Codes

Shi M., Özbudak F., Sole P.

IEEE Transactions on Information Theory, vol.67, no.6, pp.3735-3751, 2021 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 67 Issue: 6
  • Publication Date: 2021
  • Doi Number: 10.1109/tit.2021.3069772
  • Journal Name: IEEE Transactions on Information Theory
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, PASCAL, Aerospace Database, Applied Science & Technology Source, Business Source Elite, Business Source Premier, Communication Abstracts, Compendex, Computer & Applied Sciences, INSPEC, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.3735-3751
  • Keywords: Cyclic code, b-symbol error, algebraic curve, Weil-Serre bound, irreducible cyclic code, FUNCTION-FIELDS, FINITE-FIELDS, CURVES, NUMBER
  • Middle East Technical University Affiliated: Yes

Abstract

© 1963-2012 IEEE.Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2 , and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and Fq is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤n-1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q.