Group Codes

Abstract

The group codes constitute a vanishingly small percentage of all possible block codes. With very few exceptions, however, they are the only block codes of practical importance. They are often referred to by other names such as linear codes or generalized parity check codes. Within the set of all group codes there is a second major subdivision called polynomial generated codes. Examples of polynomial generated codes are the Bose-ChaudhuriHocquenghen (BCH) codes, Reed-Solomon codes, generalized Reed-Mueller codes, projective geometry codes, Euclidean geometry codes, and quadratic residue codes. Each of these classes of codes is described by a specific algorithm for constructing the code. The classes form overlapping sets so that a particular code may be a BCH code and also a residue code or it may be a generalized Reed-Mueller code and also a BCH code, etc. Polynomial generated codes are important for several reasons. First of all, the encoder can be implemented in hardware using a relatively simple feedback shift register. Second, this class contains many codes whose minimum distance is close to the best that can be found, especially for block lengths on the order of 100 or less. Third, there exist several decoding algorithms which enable one to decode certain of these codes using only moderate amounts of hardware.