Some results on ordinary words of standard Reed-Solomon codes

Abstract

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].

Keywords

• Reed-Solomon code
• ordinary word
• Konig-Rados theorem