In this chapter, the basic mathematical concepts necessary for working with non-binary error-correcting codes are presented. The elements of the non-binary codes described in this book belong to a non-binary alphabet and so this chapter begins with a discussion of the definition and properties of a Group, which will lead to an introduction to Rings and Fields. Later in the book, knowledge of the properties of rings will be necessary to understanding the design of ring trellis coded modulation (ring-TCM) codes and ring block coded modulation (ring-BCM) codes. Similarly, a good understanding of finite fields is needed to be able to construct and decode Bose, Ray-Chaudhuri, Hocquenghem (BCH) codes, Reed-Solomon codes and Algebraic-Geometric codes.
2.1.1 Groups
A set contains any elements or objects with no conditions imposed on it and can be either finite or infinite. The number of elements or objects in a set is called the cardinality. There are two binary operations that can operate on a set: multiplication 'โข' and addition '+'. Certain conditions can be applied to the set under these binary operations. The most common conditions are [1]: r Commutativity: For two elements a and b in the set, aโขb = bโขa under multiplication or a + b = b + a under addition. r Identity: For any element a in the set there is an identity element b such that aโขb = a under multiplication or a + b = a under addition.
r Inverse: For any element a in the set its inverse a -1 must also be in the set. This obeys aโขa -1 = a -1 โขa = b (the identity element).