Apart from the construction of BCH, RS, finite-geometry, and RM codes based on finite fields and finite geometries, there are other methods (or techniques) for constructing long powerful codes from good short codes.

Besides Euclidean and projective geometries, there are other types of finite geometries. One such type is known as partial geometries. Akin to Euclidean and projective geometries, partial geometries can be used to construct LDPC codes whose Tanner graphs have similar structural properties as those of Euclidean- and projective-geometry LDPC codes.

Bose–Chaudhuri–Hocquenghem (BCH) codes form a large class of cyclic codes for correcting multiple random errors. This class of codes was first discovered by Hocquenghem in 1959 [] and independently by Bose and Chaudhuri in 1960 []. The first algorithm for decoding binary BCH codes was devised by Peterson in 1960 [].

Reed–Muller (RM) codes form a class of multiple error-correcting codes. These codes were first discovered by Muller in 1954 []. In the same year, Reed devised a simple method for decoding these codes [].

As presented in , there are two categories of error-correcting codes, block codes and convolutional codes. This chapter gives an introduction to linear block codes, a subclass of block codes. The coverage of this chapter includes: (1) fundamental concepts and structures of linear block codes; (2) generation of these codes in terms of their generator and parity-check matrices; (3) their error detection and correction capabilities; and (4) general decoding of these codes. We will begin the introduction of linear block codes with symbols from the binary field GF(2). Linear block codes over nonbinary fields, which have similar structures and properties to those over the binary field GF(2), will be briefly discussed at the end of this chapter.

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.