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 5: Bose-Chaudhuri-Hocquenghem Codes
5.1 Introduction
Bose, Chaudhuri, and Hocquenghem (BCH) codes form a class of powerful random- and multiple-error-correcting cyclic codes. The codes were discovered by Bose and Chaudhuri in 1960, and by Hocquenghem, in 1959 [1 3]. They are relatively easy to encode and decode using algebraic decoding [4, 5]. Algebraic decoding is possible because the considerable mathematical structure of the codes makes it possible to find algorithms for solving the syndrome equation. The codes can be described in the time domain or in terms of the Galois-field Fourier transform; i.e., in the frequency domain. The treatment here will concentrate on the basic principles of binary BCH codes and the algebraic decoding of the codes.
5.2 General Description of BCH Codes
Let ? be an element of GF( q s) and let an integer 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 = b, b + 1,.
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