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On the covering radii of a class of binary primitive cyclic codes

Year 2022, Volume: 51 Issue: 1, 20 - 26, 14.02.2022

Seher Tutdere

https://doi.org/10.15672/hujms.881649
Cited By: 1

Abstract

 In 2019, Kavut and Tutdere proved that the covering radii of a class of primitive binary cyclic codes with minimum distance greater than or equal to $r+2$ is $r$, where $r$ is an odd integer, under some assumptions. We here show that the covering radii $R$ of a class of primitive binary cyclic codes with minimum distance strictly greater than $\ell$ satisfy $r\leq R \leq \ell$, where $\ell,r$ are some integers, with $\ell$ being odd, depending on the given code. This new class of cyclic codes covers that of Kavut and Tutdere. 

Keywords

cyclic code covering radius finite field polynomial equations

References

  • [1] S.V. Bezzateev and N.A. Shekhunova, Lower Bounds on the Covering Radius of the Non-Binary and Binary Irreducible Goppa Codes, IEEE Trans. Inform. Theory 64 (11), 7171–7177, 2018.
  • [2] G.D. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, Elsevier, 1997.
  • [3] G.D. Cohen, M.G. Karpovsky, H.F. Jr. Mattson and J.R. Schatz, Covering radius- survey and recent results, IEEE Trans. Inform. Theory 31 (3), 328–343, 1985.
  • [4] G.D. Cohen, S.N. Litsyn, A.C. Lobstein and H.F. Jr. Mattson, Covering radius 1985– 1994, Appl. Algebra Engrg. Comm. Comput. 8 (3), 173–239, 1997.
  • [5] P. Delsarte, Four fundamental parameters of a code and their combinatorial signifi- cance, Inf. Control 23, 407–438, 1973.
  • [6] D. Gorenstein, W.W. Peterson and N. Zierler, Two-error correcting Bose-Chaudhuri codes are quasi-perfect, Inf. Control 3 (3), 291–294, 1960.
  • [7] T. Helleseth, On the covering radius of cyclic linear codes and arithmetic codes, Dis- crete Appl. Math. 11.2, 157–173, 1985.
  • [8] F.T. Howard, The power of 2 dividing the coefficients of certain power series, Fi- bonacci Quart. 39 (4), 358–363, 2001.
  • [9] S. Kavut and S. Tutdere, The covering radii of a class of binary cyclic codes and some BCH codes, Des. Codes Cryptogr. 87, 317–325, 2019.
  • [10] O. Moreno and N.F. Castro, Divisibility properties for covering radius of certain cyclic codes, IEEE Trans. Inform. Theory 49 (12), 3299–3303, 2003.
  • [11] O. Moreno and C.J. Moreno, Improvement of Chevalley-Warning and the Ax-Katz Theorems, Amer. J. Math. 117 (1), 241–244, 1995.
  • [12] J.H. Van Lint and R. Wilson, On the minimum distance of cyclic codes, IEEE Trans. Inform. Theory 32 (1), 23–40, 1986.

Year 2022, Volume: 51 Issue: 1, 20 - 26, 14.02.2022

Seher Tutdere

https://doi.org/10.15672/hujms.881649
Cited By: 1

Abstract

References

  • [1] S.V. Bezzateev and N.A. Shekhunova, Lower Bounds on the Covering Radius of the Non-Binary and Binary Irreducible Goppa Codes, IEEE Trans. Inform. Theory 64 (11), 7171–7177, 2018.
  • [2] G.D. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, Elsevier, 1997.
  • [3] G.D. Cohen, M.G. Karpovsky, H.F. Jr. Mattson and J.R. Schatz, Covering radius- survey and recent results, IEEE Trans. Inform. Theory 31 (3), 328–343, 1985.
  • [4] G.D. Cohen, S.N. Litsyn, A.C. Lobstein and H.F. Jr. Mattson, Covering radius 1985– 1994, Appl. Algebra Engrg. Comm. Comput. 8 (3), 173–239, 1997.
  • [5] P. Delsarte, Four fundamental parameters of a code and their combinatorial signifi- cance, Inf. Control 23, 407–438, 1973.
  • [6] D. Gorenstein, W.W. Peterson and N. Zierler, Two-error correcting Bose-Chaudhuri codes are quasi-perfect, Inf. Control 3 (3), 291–294, 1960.
  • [7] T. Helleseth, On the covering radius of cyclic linear codes and arithmetic codes, Dis- crete Appl. Math. 11.2, 157–173, 1985.
  • [8] F.T. Howard, The power of 2 dividing the coefficients of certain power series, Fi- bonacci Quart. 39 (4), 358–363, 2001.
  • [9] S. Kavut and S. Tutdere, The covering radii of a class of binary cyclic codes and some BCH codes, Des. Codes Cryptogr. 87, 317–325, 2019.
  • [10] O. Moreno and N.F. Castro, Divisibility properties for covering radius of certain cyclic codes, IEEE Trans. Inform. Theory 49 (12), 3299–3303, 2003.
  • [11] O. Moreno and C.J. Moreno, Improvement of Chevalley-Warning and the Ax-Katz Theorems, Amer. J. Math. 117 (1), 241–244, 1995.
  • [12] J.H. Van Lint and R. Wilson, On the minimum distance of cyclic codes, IEEE Trans. Inform. Theory 32 (1), 23–40, 1986.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Seher Tutdere 0000-0001-5645-8174

Publication Date February 14, 2022
Published in Issue Year 2022 Volume: 51 Issue: 1

Cite

APA Tutdere, S. (2022). On the covering radii of a class of binary primitive cyclic codes. Hacettepe Journal of Mathematics and Statistics, 51(1), 20-26. https://doi.org/10.15672/hujms.881649
AMA Tutdere S. On the covering radii of a class of binary primitive cyclic codes. Hacettepe Journal of Mathematics and Statistics. February 2022;51(1):20-26. doi:10.15672/hujms.881649
Chicago Tutdere, Seher. “On the Covering Radii of a Class of Binary Primitive Cyclic Codes”. Hacettepe Journal of Mathematics and Statistics 51, no. 1 (February 2022): 20-26. https://doi.org/10.15672/hujms.881649.
EndNote Tutdere S (February 1, 2022) On the covering radii of a class of binary primitive cyclic codes. Hacettepe Journal of Mathematics and Statistics 51 1 20–26.
IEEE S. Tutdere, “On the covering radii of a class of binary primitive cyclic codes”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, pp. 20–26, 2022, doi: 10.15672/hujms.881649.
ISNAD Tutdere, Seher. “On the Covering Radii of a Class of Binary Primitive Cyclic Codes”. Hacettepe Journal of Mathematics and Statistics 51/1 (February2022), 20-26. https://doi.org/10.15672/hujms.881649.
JAMA Tutdere S. On the covering radii of a class of binary primitive cyclic codes. Hacettepe Journal of Mathematics and Statistics. 2022;51:20–26.
MLA Tutdere, Seher. “On the Covering Radii of a Class of Binary Primitive Cyclic Codes”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, 2022, pp. 20-26, doi:10.15672/hujms.881649.
Vancouver Tutdere S. On the covering radii of a class of binary primitive cyclic codes. Hacettepe Journal of Mathematics and Statistics. 2022;51(1):20-6.

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