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Cyclic CodesCyclic codes are a subset of linear block codes. They are the most useful andpopular because encoding and decoding can be implemented by using simpleshift registers: error detection and correction are achieved with shift registersand some additional logic gates. With a cyclic code, an end-around shift of acodeword yields another codeword. For example, ifðy 1 ; y 2 ; y 3 ; ; y n1 ; y n Þis a codeword, then, by definition, a cyclic code is also a codeword of the formðy n ; y 1 ; y 2 ; y 3 ; ; y n1 ÞIt follows from the above cyclic code description that only n 1 codewordscan be generated by cyclic shifts of a single codeword.Encoding is achieved by using either an ðn kÞ-stage shift register or ak-stage shift register. Syndrome calculation for decoding is achieved by anðn kÞ-stage shift register or a k-stage shift register. An example of encodingand decoding of a cyclic code is shown in Fig. 6.1.Fig. 6.1(a) is a (7, 4) encoder. It generates a 7-bit codeword using a 3-shift register ðr 0 ; r 1 ; r 2 Þ. The message bits form the first 4 bits of the 7-bitcodeword. When the gate is closed and the output switch is set to position 1,the most significant bits of the message bits are transmitted first. As the 4message bits are transmitted, the gate is opened, allowing the data bits to beshifted into the 3-shift register. After the 4 bits have been transmitted and fedinto the shift register, the output switch is set to position 2. The contents of theshift register are then transmitted, forming the 3 parity check bits.The output buffer from the encoder is the 7-bit-long codeword, which isfed into the syndrome forming part of the decoder in Fig. 6.1(b). Thesyndrome output is AND-gated, acting as the error detector. The AND gategives a binary ‘‘1’’ output only when the AND input equals 100. If the ANDoutput is ‘‘0,’’ the current bit in the received codeword is correct and thereceived sequence is then shifted out of the decoder.Cyclic codes have rich algebraic properties, which allow for efficientdecoding techniques. A number of books, including Peterson [11], Lin andCostello [9], and Shamnugam [10], have provided excellent mathematicaltreatment of cyclic codes that inquiring readers should consult.BCH CodesThe BCH (Bose–Chaudhuri–Hocquenghem) codes [12] are perhaps the mostpowerful and flexible group of cyclic error correction codes available. TheirCopyright © 2002 by Marcel Dekker, Inc. All Rights Reserved.