A class of cyclic codes constructed via semiprimitive two-weight irreducible cyclic codes

Year 2017, , 281 - 290, 15.09.2017

Jesús E. Cuén-ramos Gerardo Vega

https://doi.org/10.13069/jacodesmath.327385

Abstract

We present a family of reducible cyclic codes constructed as a direct sum (as vector spaces) of two different semiprimitive two-weight irreducible cyclic codes. This family generalizes the class of reducible cyclic codes that was reported in the main result of [10]. Moreover, despite of what was stated therein, we show that, at least for the codes studied here, it is still possible to compute the frequencies of their weight distributions through the cyclotomic numbers in an easy way.

Keywords

Weight distribution, Reducible cyclic codes, Semiprimitive cyclic codes, Cyclotomic numbers

References

• [1] P. Delsarte, On subfield subcodes of Reed–Solomon codes, IEEE Trans. Inform. Theory 21(5) (1975) 575–576.
• [2] C. Ding, Y. Liu, C. Ma, L. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inform. Theory 57(12) (2011) 8000–8006.
• [3] T. Helleseth, Some two–weight codes with composite parity–check polynomials, IEEE Trans. Inform. Theory 22(5) (1976) 631–632.
• [4] R. Lidl, H. Niederreiter, Finite Fields, Cambridge Univ. Press, Cambridge 1983.
• [5] C. Ma, L. Zeng, Y. Liu, D. Feng, C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inform. Theory 57(1) (2011) 397–402.
• [6] G. Vega, Two–weight cyclic codes constructed as the direct sum of two one–weight cyclic codes, Finite Fields Appl. 14(3) (2008) 785–797.
• [7] G. Vega, The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inform. Theory 58(7) (2012) 4862–4869.
• [8] G. Vega, L. B. Morales, A general description for the weight distribution of some reducible cyclic codes, IEEE Trans. Inform. Theory 59(9) (2013) 5994–6001.
• [9] G. Vega, A critical review and some remarks about one– and two–weight irreducible cyclic codes, Finite Fields Appl. 33 (2015) 1–13.
• [10] B. Wang, C. Tang, Y. Qi, Y. Yang, M. Xu, The weight distributions of cyclic codes and elliptic curves, IEEE Trans. Inform. Theory 58(12) (2012) 7253–7259.
• [11] J. Wolfmann, Are 2-weight projective cyclic codes irreducible?, IEEE Trans. Inform. Theory 51(2) (2005) 733–737.
• [12] M. Xiong, The weight distributions of a class of cyclic codes, Finite Fields Appl. 18(5) (2012) 933–945.
• [13] 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.