Error-Control Block Codes for Communications Engineers
An engineer s guide to the theory behind the common random-error-control block codes, and their applications in digital communications.
TABLE OF CONTENTS Error-Control Block Codes for Communications Engineers
Chapter 6: Reed-Solomon Codes
6.1 Introduction
Reed-Solomon (RS) codes form a subclass of the nonbinary BCH codes. The Reed-Solomon codes are cyclic codes and the codes were discovered by Reed and Solomon in 1960 [1]. Although they are a subclass of the nonbinary BCH codes, Reed-Solomon codes offer better error control performance and more efficient practical implementation because they have the largest minimum Hamming distance for fixed values of k and n. In this chapter, we describe the generation and decoding of Reed-Solomon codes. A thorough discussion of the Reed-Solomon codes can be found in references [2 6].
6.2 Description of Reed-Solomon Codes
Let ? be an element of GF( q s) and let t d be the designed error-correcting power of a BCH code. In Chapter 5, we have seen that for some positive integers s and b ? 1, a BCH code of length n and minimum Hamming distance ? 2 t d + 1 can be generated by the generator polynomial g( x) over GF( q) with ? b, ? b+1, . . . , ? b+2 t d ?1 as the roots of g( x). Let ? i, a nonzero element in GF( q s), be a root of the minimal polynomial ? i( x) over GF( q) and n i be the order of ? i, for i
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