rs can be alleviated pendent on two things: (1) the decoder that is used to with interleaving. decode a received word to a codeword and (2) the distance between each pair of codewords in the block code. Let us 3.4.3.1 Linear block codes first consider the distance between a pair of codewords. For block codes, the Hamming metric or the Euclidean In a linear block code, the codeword symbols are taken metric is usually used to measure the distance between from a field. A formal definition of a field is beyond the pairs of codewords. The Hamming distance is useful for scope of this text and may be found in a mathematics measuring the distance between two codewords whose book on abstract algebra (see [12]). Informally, a field symbols belong to a finite field. Sometimes in communi- consists of a set of elements, together with two operations cations applications, however, each field element of the called addition and multiplication that must fulfill a given codeword is mapped to a real number. In such scenarios, it number of properties. Some examples of well-known is useful to use a Euclidean metric to determine the fields are the set of real numbers and the set of rational distance between codewords. For example, if each numbers. These fields are known as infinite fields because codeword, Ci = (Ci,1, ci,n), is mapped to a vector of they contain an infinite number of elements. Linear block real numbers, ri = ,riin), then the Euclidean codes, however, typically consist of elements from a finite distance between two codewords Ci, Cj may be defined as field. In particular, consider the finite field GF(2) = {0,1}. The addition operation for GF(2) is modulo-2 addition de(ci,cj) = and the multiplication operation is defined similarly to the multiplication operation for two binary numbers: (3.4.3) 0+0 = 0 + 1 = = 0 = Now, if we let dmin represent the minimum distance between any pair of codewords, and if an arbitrary codeword is transmitted over a noisy channel, then the Finite fields are also called Galois fields. The size o