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Using a block code for error control, each codeword is generated and transmitted independently. At the receiving end, each received vector is decoded independently without using the reliability information of previously decoded received vectors.
Right after the discovery of binary BCH codes by Hocquenghem in 1959 [] and by Bose and Chaudhuri in 1960 [] independently, Gorenstein and Zierler extended this class of codes to the nonbinary case in 1961 [].
Low-density parity-check (LDPC) codes form a class of linear block codes whose parity-check matrices are low-density (or sparse). This class of codes can achieve near-capacity (or near Shannon limit) [] performance on various communication and data-storage channels. LDPC codes were discovered by Gallager in 1962 [, ].
This chapter presents a simple model of a digital communication (or storage) system and its key function units, which are relevant for reliable information transmission (or storage) over a transmission (or storage) medium subject to noise disturbance (or medium defect). The two function units that play the key role in protection of transmitted (or stored) information against noise (or medium defect) are encoder and decoder. The function of the encoder is to transform an information sequence into another sequence, called a coded sequence, which enables the detection and correction of transmission errors at the receiving end of the system.
Inand , two types of cyclic codes, namely, BCH and RS codes, are constructed based on finite fields, and they not only have distinctive algebraic structures but are also powerful in correcting errors. In this chapter, we show that cyclic codes can be constructed based on finite geometries.
showed that cyclic codes can be constructed based on the lines of two classes of finite geometries, namely, Euclidean and projective geometries. It was shown that these cyclic finite-geometry (FG) codes can be decoded with the simple one-step majority-logic decoding (OSMLD) based on the orthogonal structure of their parity-check matrices.
In the previous five chapters, we have been primarily concerned with codes and coding techniques for channels on which transmission errors occur independently, i.e., random errors. However, there are communication and data-storage systems where errors tend to be localized in nature.
The objective of this chapter is to present some important elements of modern algebra and graphs that are pertinent to the understanding of the fundamental structural properties and constructions of some important classes of classical and modern error-correcting codes. The elements to be covered are groups, finite fields, polynomials, vector spaces, matrices, and graphs.