On Ordinary Words of Standard Reed-Solomon Codes over Finite Fields

• 作者: Xiaofan Xu Shaofang Hong
• 来源期刊: 代数集刊（英文版）
• 年，卷（期）: 2021,（4）
• 页码范围: 569-580
• 页数: 12页
• 语种: 英文
• DOI: 10.1142/S1005386721000456

摘要

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.Usually we use the maximum likelihood decoding(MLD) algorithm in the decoding process of Reed-Solomon codes.MLD algorithm relies on determining the error distance of received word.Dür,Guruswami,Wan,Li,Hong,Wu,Yue and Zhu et al.got some results on the error distance.For the Reed-Solomon code C,the received word u is called an ordinary word of C if the error distance d(u,C) =n-deg u(x) with u(x) being the Lagrange interpolation polynomial of u.We introduce a new method of studying the ordinary words.In fact,we make use of the result obtained by Y.C.Xu and S.F.Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed-Solomon codes over the finite field of q elements.This completely answers an open problem raised by Li and Wan in[On the subset sum problem over finite fields,Finite Fields Appl.14 (2008) 911-929].