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Figure 14.1 A typical transmission system with channel coding
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A simple technique called automatic request for retransmission scheme can be used in error control applications. In this method, the transmitter stops and waits until a correct receipt of the data is acknowledged, or a request for retransmission is received on the backward channel. The request for retransmission can be simply based on a CRC method.
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14.2 Block Codes
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A block code [2] consists of a set of xed-length vectors called codewords. The length n of a codeword is the number of elements in the vector. The elements of a codeword are selected from an alphabet of q elements. When the alphabet consists of two elements 0 and 1, the code is a binary code, and these elements are called bits. The Galois eld (GF) arithmetic is a nite state eld, and GF(q) is Galois eld of order q. The nite eld with two elements is denoted as GF(2). When the elements of a codeword are selected from an alphabet having q elements where q > 2, the code is a nonbinary code. When q is a power of 2 (i.e., q = 2m where m is a positive integer), each q-ary element has an equivalent binary representation consisting of m bits. Thus, a nonbinary code of block length N can be mapped into a binary code of block length n = mN. This nonbinary nite eld is denoted by GF(q) or GF(2m ) where m > 1. There are 2n possible codewords in a binary block code of length n. From these 2n codewords, we may select M = 2k codewords (k < n) to form a code. Thus, a block of k information bits is mapped into a codeword of length n selected from the set of M = 2k codewords. The resulting block code is called an (n, k) code, and the code rate is Rc = k/n. An (n, k) Reed Solomon (RS) code is capable of correcting t symbol errors where t = (n k)/2. The block length of standard Reed Soloman codes is n = q 1, and the code rate is Rc = k/n. Examples of codes with different values of m, q, n, and t are summarized in Table 14.1. The family of linear block codes is illustrated in Figure 14.2. The cyclic, BCH (Bose Chaudhuri Hocquenghem), Hamming, and Reed Solomon codes are special classes of linear block codes: Linear block codes [4]: Suppose C1 and C2 are two codewords in an (n, k) block code. Let 1 and 2 be any two elements selected from the alphabet. The code is linear if and only if 1 C1 + 2 C2 is also a codeword.
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Table 14.1 M 6 8 8 8 8 q = 2m 64 256 256 256 256 n (Block length) 63 219 28 32 255
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Examples of Reed Solomon codes 2t(Parity symbols) 6 22 4 4 22 Rc = k/n 47/63 201/219 24/28 28/32 233/255 Applications U.S. cellular digital packet data Intelsat IESS310 Compact CD C1 encoder Compact CD C2 encoder Infrared wireless audio
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k (Information symbols) 47 201 24 28 233
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Relationship of different block codes
Cyclic codes: A cyclic code is a linear block code with the property that a cyclic shift of a codeword is also a codeword. For example, if C = [cn 1 , cn 2 , . . . , c1 , c0 ] is a codeword of a cyclic code, then [cn 2 , . . . , c1 , c0 , cn 1 ] obtained from the cyclic shift of the element C is also a codeword. This cyclic property means cyclic code possesses a considerable amount of structures that can be exploited in the encoding and decoding operations. An (n, k) cyclic code is completely speci ed by the following generator polynomial: g(x) = 1 + g1 x + + gn k 1 x n k 1 + gn k x n k . (14.2)
BCH codes [5]: The BCH code is a class of cyclic codes whose generator polynomial is the product of distinct minimal polynomials corresponding to , 2 , . . . , 2t , where G F(2m ) is the root of the primitive polynomial p(x). The most important and common class of nonbinary BCH codes is the Reed Solomon codes. Figure 14.3 shows a Reed Solomon codeword in which the data is unchanged while the parity bits are suf xed to the data bits. The Reed Solomon codes are the most commonly used for practical applications. Reed Solomon codes use nonbinary elds GF(2m ). These elds have more than two elements and are extensions of the binary eld GF(2) = {0, 1}. The additional elements in the extension eld use a new symbol to represent the elements other than 0 and 1. Each nonzero element can be represented by a power of . Some important Galois eld properties are: 1. An element with order (q 1) in GF(q) is called a primitive element. Each eld contains at least one primitive element . All nonzero elements in GF(q) can be represented as (q 1) consecutive powers of a primitive element .