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

Year 2019, , 1 - 11, 19.01.2019

Manjit Singh , Sudhir Batra

https://doi.org/10.13069/jacodesmath.505364

Cited By: 1

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 $q\equiv3\pmod 4$.

Keywords

Linear codes, Reversible codes, Weight distributions, Constant weight codes

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.