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Weight distribution of a class of cyclic codes of length $2^n$

Year 2019, , 1 - 11, 19.01.2019
https://doi.org/10.13069/jacodesmath.505364

Abstract

Let $\mathbb{F}_q$ be a finite field with $q$ elements and $n$ be a positive integer. In this paper, we determine the weight distribution of a class cyclic codes of length $2^n$ over $\mathbb{F}_q$ whose parity check polynomials are either binomials or trinomials with $2^l$ zeros over $\mathbb{F}_q$, where integer $l\ge 1$. In addition, constant weight and two-weight linear codes are constructed when $q\equiv3\pmod 4$.

References

  • [1] E. R. Berlekamp, Algebraic Coding Theory, Revised Edition, World Scientific Publishing Co. Pte. Ltd., 2015.
  • [2] C. Ding, D. R. Kohel, S. Ling, Secret–sharing with a class of ternary codes, Theor. Comput. Sci. 246(1–2) (2000) 285–298.
  • [3] H. Q. Dinh, C. Li, Q. Yue, Recent progress on weight distributions of cyclic codes over finite fields, J. Algebra Comb. Discrete Struct. Appl. 2(1) (2015) 39–63.
  • [4] K. Feng, J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl. 14(2) (2008) 390–409.
  • [5] W. C. Huffman, V. Pless, Fundamentals of Error–Correcting Codes, Cambridge University Press, Cambridge, 2003.
  • [6] A. Kathuria, S. K. Arora, S. Batra, On traceability property of equidistant codes, Discrete Math. 340(4) (2017) 713–721.
  • [7] R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, Cambridge, 1986.
  • [8] J. L. Massey, Reversible codes, Inform. Control 7(3) (1964) 369–380.
  • [9] A. Sharma, G. K. Bakshi, M. Raka, The weight distributions of irreducible cyclic codes of length $2^m$, Finite Fields Appl. 13(4) (2007) 1086–1095.
  • [10] M. Singh, S. Batra, Some special cyclic codes of length $2^n$, J. Algebra Appl. 16(1) (2017) 17 pages.
  • [11] G. Vega, The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inform. Theory, 58(7) (2012) 4862–4869.
  • [12] M. Van Der Vlugt, Hasse–Davenport curves, Gauss sums and weight distributions of irreducible cyclic codes, J. Number Theory 55(2) (1995) 145–159.
  • [13] Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, Singapore, 2003.
  • [14] J. Yang, M. Xiong, C. Ding, J. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inform. Theory 59(9) (2013) 5985–5993.
  • [15] X. Zhu, Q. Yue, L. Hu, Weight distributions of cyclic codes of length $l^m$, Finite Fields Appl. 31 (2015) 241–257.
Year 2019, , 1 - 11, 19.01.2019
https://doi.org/10.13069/jacodesmath.505364

Abstract

References

  • [1] E. R. Berlekamp, Algebraic Coding Theory, Revised Edition, World Scientific Publishing Co. Pte. Ltd., 2015.
  • [2] C. Ding, D. R. Kohel, S. Ling, Secret–sharing with a class of ternary codes, Theor. Comput. Sci. 246(1–2) (2000) 285–298.
  • [3] H. Q. Dinh, C. Li, Q. Yue, Recent progress on weight distributions of cyclic codes over finite fields, J. Algebra Comb. Discrete Struct. Appl. 2(1) (2015) 39–63.
  • [4] K. Feng, J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl. 14(2) (2008) 390–409.
  • [5] W. C. Huffman, V. Pless, Fundamentals of Error–Correcting Codes, Cambridge University Press, Cambridge, 2003.
  • [6] A. Kathuria, S. K. Arora, S. Batra, On traceability property of equidistant codes, Discrete Math. 340(4) (2017) 713–721.
  • [7] R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, Cambridge, 1986.
  • [8] J. L. Massey, Reversible codes, Inform. Control 7(3) (1964) 369–380.
  • [9] A. Sharma, G. K. Bakshi, M. Raka, The weight distributions of irreducible cyclic codes of length $2^m$, Finite Fields Appl. 13(4) (2007) 1086–1095.
  • [10] M. Singh, S. Batra, Some special cyclic codes of length $2^n$, J. Algebra Appl. 16(1) (2017) 17 pages.
  • [11] G. Vega, The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inform. Theory, 58(7) (2012) 4862–4869.
  • [12] M. Van Der Vlugt, Hasse–Davenport curves, Gauss sums and weight distributions of irreducible cyclic codes, J. Number Theory 55(2) (1995) 145–159.
  • [13] Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, Singapore, 2003.
  • [14] J. Yang, M. Xiong, C. Ding, J. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inform. Theory 59(9) (2013) 5985–5993.
  • [15] X. Zhu, Q. Yue, L. Hu, Weight distributions of cyclic codes of length $l^m$, Finite Fields Appl. 31 (2015) 241–257.
There are 15 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Manjit Singh 0000-0003-3351-7287

Sudhir Batra This is me 0000-0003-4139-0589

Publication Date January 19, 2019
Published in Issue Year 2019

Cite

APA Singh, M., & Batra, S. (2019). Weight distribution of a class of cyclic codes of length $2^n$. Journal of Algebra Combinatorics Discrete Structures and Applications, 6(1), 1-11. https://doi.org/10.13069/jacodesmath.505364
AMA Singh M, Batra S. Weight distribution of a class of cyclic codes of length $2^n$. Journal of Algebra Combinatorics Discrete Structures and Applications. January 2019;6(1):1-11. doi:10.13069/jacodesmath.505364
Chicago Singh, Manjit, and Sudhir Batra. “Weight Distribution of a Class of Cyclic Codes of Length $2^n$”. Journal of Algebra Combinatorics Discrete Structures and Applications 6, no. 1 (January 2019): 1-11. https://doi.org/10.13069/jacodesmath.505364.
EndNote Singh M, Batra S (January 1, 2019) Weight distribution of a class of cyclic codes of length $2^n$. Journal of Algebra Combinatorics Discrete Structures and Applications 6 1 1–11.
IEEE M. Singh and S. Batra, “Weight distribution of a class of cyclic codes of length $2^n$”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 6, no. 1, pp. 1–11, 2019, doi: 10.13069/jacodesmath.505364.
ISNAD Singh, Manjit - Batra, Sudhir. “Weight Distribution of a Class of Cyclic Codes of Length $2^n$”. Journal of Algebra Combinatorics Discrete Structures and Applications 6/1 (January 2019), 1-11. https://doi.org/10.13069/jacodesmath.505364.
JAMA Singh M, Batra S. Weight distribution of a class of cyclic codes of length $2^n$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2019;6:1–11.
MLA Singh, Manjit and Sudhir Batra. “Weight Distribution of a Class of Cyclic Codes of Length $2^n$”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 6, no. 1, 2019, pp. 1-11, doi:10.13069/jacodesmath.505364.
Vancouver Singh M, Batra S. Weight distribution of a class of cyclic codes of length $2^n$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2019;6(1):1-11.

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