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Dergiler / Journal of Algebra Combinatorics Discrete Structures and Applications / 2015 / Cilt: 2 - Sayı: 1

Recent progress on weight distributions of cyclic codes over finite fields

Hai Q. Dinh
Chengju Li
Qin Yue
Sayfa
39–63
Yayın Tarihi
2015-01-22
DOI
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Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms.
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Temel Bulgular & Veri:In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability.
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Sonuç & Katkı:Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Linear codes · Weight distribution · Cyclic codes

Özet

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions.

Abstract

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions.

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Bu Makaleye Atıf Yap

Dinh, H. Q., Li, C., Yue, Q. (2015). Recent progress on weight distributions of cyclic codes over finite fields. Journal of Algebra Combinatorics Discrete Structures and Applications, Cilt: 2 - Sayı: 1, 39–63. https://ajindex.com/articles/861714

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Kaynakça (15)

  1. N. Aydin, I. Siap, D. K. Ray-Chaudhuri, The structure of 1-generator quasi-twisted codes and new linear codes, Designs Codes Cryptogr. 24, 313-326, 2001.
  2. A. Batoul, K. Guenda, T. A. Gulliver, On self-dual cyclic codes over finite chain rings, Designs Codes Cryptogr. 70, 347-358, 2014.
  3. E. R. Berlekamp, Negacyclic Codes for the Lee Metric, Proceedings of the Conference on Combi natorial Mathematics and Its Applications, Chapel Hill, N.C., University of North Carolina Press, 298-316, 1968.
  4. E. R. Berlekamp, Algebraic Coding Theory, revised 1984 edition, Aegean Park Press, 1984.
  5. S. D. Berman, Semisimple cyclic and Abelian codes. II, Kibernetika (Kiev) 3 (1967), 21-30 (Russian). English translation: Cybernetics 3, 17-23,1967.
  6. B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi Sums, Wiley-Interscience Publication, 1998.
  7. C. Berrou, A. Glavieux, and P. Thitimajshima, Near Shannon limit error-correcting coding and decoding: Turbo-codes, 1993.
  8. A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. A. Kohnert, A. Wassermann, Error Correcting Linear Codes: Classification by Isometry and Applications, Springer, Berlin, 2006.
  9. N. Boston and G. McGuire, The weight distribution of cyclic codes with two zeros and zeta functions, J. Symbolic Comput., 45(7), 723-733, 2010.
  10. A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs, Berlin, Germany: Springer- Verlag, 18, 1989, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. A. R. Calderbank and J. M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39, 143-152, 1984.
  11. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18, A. Canteaut, P. Charpin, and H. Dobbertin, Weight divisibility of cyclic codes, highly nonlinear functions on F2mand crosscorrelation of maximum-length sequences, SIAM J. Discrete Math., 13(1), 105-138, 2000.
  12. C. Carlet, P. Charpin, and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr. 15(2), 125-156, 1998.
  13. C. Carlet and C. Ding, Highly nonlinear mappings, J. Complexity, 20(2), 205-244, 2004.
  14. G. Castagnoli, J. L. Massey, P. A. Schoeller and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inform. Theory 37, 337-342, 1991.
  15. P. Charpin, Open problems on cyclic codes, in Handbook of Coding Theory, Part 1: Algebraic Coding, V. S. Pless, W. C. Huffman, and R. A. Brualdi, Eds. Amsterdam, The Netherlands: Elsevier, ch. 11,