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

Manjit Singh, Sudhir Batra

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 $q\equiv3\pmod 4$.

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References


E. R. Berlekamp, Algebraic Coding Theory, Revised Edition, World Scientific Publishing Co. Pte. Ltd., 2015.

C. Ding, D. R. Kohel, S. Ling, Secret–sharing with a class of ternary codes, Theor. Comput. Sci. 246(1–2) (2000) 285–298.

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.

K. Feng, J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl. 14(2) (2008) 390–409.

W. C. Huffman, V. Pless, Fundamentals of Error–Correcting Codes, Cambridge University Press, Cambridge, 2003.

A. Kathuria, S. K. Arora, S. Batra, On traceability property of equidistant codes, Discrete Math. 340(4) (2017) 713–721.

R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, Cambridge, 1986.

J. L. Massey, Reversible codes, Inform. Control 7(3) (1964) 369–380.

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.

M. Singh, S. Batra, Some special cyclic codes of length $2^n$, J. Algebra Appl. 16(1) (2017) 17 pages.

G. Vega, The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inform. Theory, 58(7) (2012) 4862–4869.

M. Van Der Vlugt, Hasse–Davenport curves, Gauss sums and weight distributions of irreducible cyclic codes, J. Number Theory 55(2) (1995) 145–159.

Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, Singapore, 2003.

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.

X. Zhu, Q. Yue, L. Hu, Weight distributions of cyclic codes of length $l^m$, Finite Fields Appl. 31 (2015) 241–257.


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