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Unformatted text preview: 1.Introduction: Cyclic codes form an important subclass of linear codes. These codes are attractive for two reasons: first, encoding and decoding can be implemented easily by employing shift-registers with feedback connections (as will be seen later); and second, because they have considerable inherent algebric structure, it is possible to find various practical methods for decoding them. In this section the reader can find a general overview of binary cyclic codes followed by their mathematical representation and their main algebric properties (section 2); The third section deals with the encoding procedure and finally, the structure of the generator and parity-check matrices will be shown. In order to fully understand the cyclic codes material, it is advised to comprehend thoroughly the linear block codes issue first. 2.Description of cyclic codes: 2.1 Cyclic code definition: Lets start with the definition of cyclic shift : If the components of an n-tuple are cyclically shifted one place to the right, we obtain another n-tuple: which is called a cyclic shift of . If the components of are cyclically shifted i places to the right, the resultant n-tuple would be : . Clearly, cyclically shifting i places to the right is equivalent to cyclically shifting n-i places to the left. Cyclic code definition: An (n,k) linear code C is called a cyclic code if every cyclic shift of a code vector in C is also a code vector in C. 2.2 Polynomial representation: In order to develop the algebric properties of a cyclic code, we treat the components of a code vector as the coefficients of a polynomial as follows: . Thus, each code vector corresponds to a polynomial of degree n-1 or less. We shall call the code polynomial of . Before looking further into cyclic codes, we'll go over some polynomial basics first:...
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- Spring '09
- Coding theory, Linear code, Cyclic code