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Unformatted text preview: An overview on error correcting codes Linear codes Cyclic codes Encoding and decoding with cyclic codes An introduction to cyclic codes Emanuele Betti 1 , Emmanuela Orsini 2 1 bettie@posso.dm.unipi.it Department of Mathematics, University of Florence, Italy 1 orsini@posso.dm.unipi.it Department of Mathematics, University of Milan, Italy May 1, 2006 An overview on error correcting codes Linear codes Cyclic codes Encoding and decoding with cyclic codes 1 An overview on error correcting codes 2 Linear codes Definitions An example Hamming distance Paritycheck and decoding 3 Cyclic codes Definitions Algebraic structure Roots of unity and defining set An example Paritycheck matrix BCH codes 4 Encoding and decoding with cyclic codes Polynomial encoding Decoding References An overview on error correcting codes Linear codes Cyclic codes Encoding and decoding with cyclic codes A communication schema B A SOURCE DESTINATION INFORMATION An overview on error correcting codes Linear codes Cyclic codes Encoding and decoding with cyclic codes A communication schema MESSAGE B A SOURCE DESTINATION TRASMITTER INFORMATION An overview on error correcting codes Linear codes Cyclic codes Encoding and decoding with cyclic codes A communication schema MESSAGE B A SOURCE DESTINATION RECEIVER TRASMITTER SIGNAL RECEIVED SIGNAL INFORMATION MESSAGE An overview on error correcting codes Linear codes Cyclic codes Encoding and decoding with cyclic codes A communication schema MESSAGE B A SOURCE DESTINATION RECEIVER NOISE SOURCE TRASMITTER SIGNAL RECEIVED SIGNAL INFORMATION MESSAGE An overview on error correcting codes Linear codes Cyclic codes Encoding and decoding with cyclic codes A communication schema MESSAGE B A SOURCE DESTINATION RECEIVER NOISE SOURCE TRASMITTER SIGNAL RECEIVED SIGNAL INFORMATION MESSAGE PROCEDURE CODING PROCEDURE DECODING CODE An overview on error correcting codes Linear codes Cyclic codes Encoding and decoding with cyclic codes Definitions A Definition for Linear Codes Let q N such that q = p k , p prime. We denote with F q the field with q elements. ( F q ) n forms an n dimensional vector space over F q . Definition A linear code C is a kdimensional vector subspace of ( F q ) n , (0 k n ). The elements of a code are called words . n is called the length and k is called the dimension . C = { v ( F q ) n  h v , c i = 0 c C } Is called the dual code of C . An overview on error correcting codes Linear codes Cyclic codes Encoding and decoding with cyclic codes Definitions A Definition for Linear Codes Let q N such that q = p k , p prime. We denote with F q the field with q elements. ( F q ) n forms an n dimensional vector space over F q ....
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 Spring '09
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