nach oben

Designs, Codes and Cryptography

Erschienen in:

31.07.2018

The covering radii of a class of binary cyclic codes and some BCH codes

verfasst von: Selçuk Kavut, Seher Tutdere

Erschienen in: Designs, Codes and Cryptography | Ausgabe 2-3/2019

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In 2003, Moreno and Castro proved that the covering radius of a class of primitive cyclic codes over the finite field \(\mathbb {F}_2\) having minimum distance 5 (resp. 7) is 3 (resp. 5). We here give a generalization of this result as follows: the covering radius of a class of primitive cyclic codes over \(\mathbb {F}_2\) with minimum distance greater than or equal to \(r+2\) is r, where r is any odd integer. Moreover, we prove that the primitive binary e-error correcting BCH codes of length \(2^f-1\) have covering radii \(2e-1\) for an improved lower bound of f.

Vorheriger Artikel On spherical codes with inner products in a prescribed interval

Nächster Artikel Combinatorial metrics: MacWilliams-type identities, isometries and extension property

Cohen D.S.: The lenght of primitive BCH codes with minimal covering radius. Des. Codes Cryptogr. 10, 5–16 (1997).MathSciNetCrossRefMATH

Cohen D.G., Karpovsky M.G., Mattson Jr. H.F., Schatz J.R.: Covering radius-survey and recent results. IEEE Trans. Inf. Theory 31(3), 328–343 (1985).MathSciNetCrossRefMATH

Cohen G.D., Litsyn S.N., Lobstein A.C., Mattson Jr. H.F.: Covering radius 1985–1994. Appl. Algebra Eng. Commun. Comput. 8(3), 173–239 (1997).MathSciNetCrossRefMATH

Delsarte P.: Four fundamental parameters of a code and their combinatorial significance. Inf. Control. 23, 407–438 (1973).MathSciNetCrossRefMATH

Gorenstein D., Peterson W.W., Zierler N.: Two-error correcting Bose–Chaudhuri codes are quasi-perfect. Inf. Control. 3(3), 291–294 (1960).MathSciNetCrossRefMATH

Helleseth T.: All binary 3-error correcting BCH codes of lenght \(2^m-1\) having covering radius \(5\). IEEE Trans. Inf. Theory 24, 257–258 (1978).CrossRefMATH

Helleseth T.: On the covering radius of cyclic linear codes and arithmetic codes. Discret. Appl. Math. 11(2), 157–173 (1985).MathSciNetCrossRefMATH

Howard F.T.: The power of 2 dividing the coefficients of certain power series. Fibonacci Q. 39(4), 358–363 (2001).MathSciNetMATH

Levy-dit-Vehel F., Litsyn S.: More on the covering radius of BCH codes. IEEE Trans. Inf. Theory 42, 1023–1028 (1996).MathSciNetCrossRefMATH

10.

Moreno O., Castro N.F.: Divisibility properties for covering radius of certain cyclic codes. IEEE Trans. Inf. Theory 49(12), 3299–3303 (2003).MathSciNetCrossRefMATH

11.

Moreno O., Moreno C.J.: Improvement of Chevalley–Warning and the Ax–Katz Theorems. Am. J. Math. 117(1), 241–244 (1995).MathSciNetCrossRefMATH

12.

Peterson W.W., Weldon E.J.: Error-Correcting Codes. MIT Press, Cambridge (1972).MATH

13.

Skorobogatov A.N.: On the covering radius of BCH codes. In: Proc. Third Soviet-Swedish International Workshop on Information Theory, pp. 308–309 (1987).

14.

Tietäväinen A.: An asymptotic bound on the covering radii of binary BCH codes. IEEE Trans. Inf. Theory 36, 211–213 (1990).MathSciNetCrossRefMATH

15.

Tietäväinen A.: On the covering radius of long binary BCH codes. Discret. Appl. Math. 16(1), 75–77 (1987).MathSciNetCrossRefMATH

16.

Van der Horst J., Berger T.: Complete decoding of triple-error-correcting binary BCH codes. IEEE Trans. Inf. Theory 22(2), 138–147 (1976).MathSciNetCrossRefMATH

17.

Van Lint J.H., Wilson R.: On the minimum distance of cyclic codes. IEEE Trans. Inf. Theory 32(1), 23–40 (1986).MathSciNetCrossRefMATH

18.

Vlãduts S.G., Skorobogatov A.N.: Covering radius for long BCH codes. Problemy Peredachi Informatsii 25, 38–45 (1989). Translated in: Probl. Inf. Transm. 25(1), 28–34 (1989).

19.

Williams F.M., Sloane N.J.: The Theory of Error-Correcting Codes. Bell Laboratories, Murray Hill (1977).

Titel: The covering radii of a class of binary cyclic codes and some BCH codes
verfasst von: Selçuk Kavut
Seher Tutdere
Publikationsdatum: 31.07.2018
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 2-3/2019
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-018-0525-y

Springer Professional

The covering radii of a class of binary cyclic codes and some BCH codes

Abstract

Premium Partner

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Weitere Artikel der Ausgabe 2-3/2019

Duplication-correcting codes

Improved decoding and error floor analysis of staircase codes

Improved user-private information retrieval via finite geometry

Linear codes from weakly regular plateaued functions and their secret sharing schemes

A new lower bound for the smallest complete (k, n)-arc in

Improved syndrome decoding of lifted -interleaved Gabidulin codes

Premium Partner