AIMS Mathematics, 2019, 4(5): 1336-1347. doi: 10.3934/math.2019.5.1336.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Some results on ordinary words of standard Reed-Solomon codes

1 Mathematical College, Sichuan University, Chengdu 610064, P. R. China;
2 Department of Mathematics, Sichuan Tourism University, Chengdu 610100, P. R. China

The Reed-Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm.We usually use the maximum likelihood decoding algorithm (MLD) in the decoding process of Reed-Solomon codes.MLD algorithm lies in determining its error distance.Li,Wan,Hong and Wu et al obtained some results on the error distance.For the ReedSolomon code $RS_q({\mathbb F}_q^*, k)$,the received word u is called an ordinary word of $RS_q({\mathbb F}_q^*, k)$,k) if the error distance $d({ u}, RS_q({\mathbb F}_q^*, k))=n-\deg(u(x))$ with u (x) being the Lagrange interpolation polynomial of u.In this paper,we make use of the polynomial method and particularly,we use the König-Rados theorem on the number of nonzero solutions of polynomial equation over finite fields to show that if $q\geq 4, 2\leq{k}\leq{q-2}$,then the received word ${ u}\in{\mathbb F}_q^{q-1}$ of degree q-2 is an ordinary word of $RS_q({\mathbb F}_q^*, k)$ if and only if its Lagrange interpolation polynomial u (x) is of the form
$$u(x)=\lambda\sum\limits_{i=k}^{q-2}a^{q-2-i}x^i+f_{\leq k-1}(x)$$
with $a, \lambda\in{\mathbb F}_q^*$ and $ f_{\leq k-1}(x)\in {\mathbb F}_q[x]$ being of degree at most k -1.This answers partially an open problem proposed by J.Y.Li and D.Q.Wan in [On the subset sum problem over finite fields,Finite Fields Appls.14(2008),911-929].
  Figure/Table
  Supplementary
  Article Metrics

Keywords Reed-Solomon code; ordinary word; Konig-Rados theorem

Citation: Xiaofan Xu, Yongchao Xu, Shaofang Hong. Some results on ordinary words of standard Reed-Solomon codes. AIMS Mathematics, 2019, 4(5): 1336-1347. doi: 10.3934/math.2019.5.1336

References

  • 1. Q. Cheng and E. Murray, On deciding deep holes of Reed-Solomon codes. In:J.Y. Cai, S.B. Cooper, H. Zhu(eds) Theory and Applications of Models of Computation. TAMC 2007, Lecture Notes in Computer Science, vol. 4484, Springer, Berlin, Heidelberg.
  • 2. S. F. Hong and R. J. Wu, On deep holes of generalized Reed-Solomon codes, AIMS Math., 1 (2016), 96-101.    
  • 3. J. Y. Li and D. Q. Wan, On the subset sum problem over finite fields, Finite Fields Th. App., 14 (2008), 911-929.    
  • 4. Y. J. Li and D. Q. Wan, On error distance of Reed-Solomon codes, Sci. China Math., 51 (2008), 1982-1988.    
  • 5. Y. J. Li and G. Z. Zhu, On the error distance of extended Reed-Solomon codes, Adv. Math. Commun., 10 (2016), 413-427.    
  • 6. R. Lidl and H. Niederreiter, Finite fields, Encyclopedia of Mathematics and its Applications, 2 Eds., Cambridge:Cambridge University Press, 1997.
  • 7. G. Rados, Zur Theorie der Congruenzen höheren Grades, J. reine angew. Math., 99 (1886), 258-260.
  • 8. G. Raussnitz, Zur Theorie de Conguenzen höheren Grades, Math. Naturw. Ber. Ungarn., 1 (1882/83), 266-278.
  • 9. R. J. Wu and S. F. Hong, On deep holes of standard Reed-Solomon codes, Sci. China Math., 55 (2012), 2447-2455.    
  • 10. X. F. Xu, S. F. Hong and Y. C. Xu, On deep holes of primitive projective Reed-Solomon codes, SCIENTIA SINICA Math., 48 (2018), 1087-1094.    
  • 11. X. F. Xu and Y. C. Xu, Some results on deep holes of generalized projective Reed-Solomon codes, AIMS Math., 4 (2019), 176-192.    
  • 12. G. Z. Zhu and D. Q. Wan. Computing error distance of Reed-Solomon codes. In:TAMC 2012 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation, (2012), 214-224.

 

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved