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 Reed-Solomon code $RS_q({\mathbb F}_q^*, k)$, the received word ${\bf u}$ is called an ordinary word of $RS_q({\mathbb F}_q^*, k)$ if the error distance $d({\bf u}, RS_q({\mathbb F}_q^*, k)) = n-\deg(u(x))$ with $u(x)$ being the Lagrange interpolation polynomial of ${\bf u}$. In this paper, we make use of the polynomial method and particularly, we use the KŚnig-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 ${\bf u}\in{\mathbb F}_q^{q-1}$ 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].