talk_betti_orsini - An overview on error correcting codes...

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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 [email protected] Department of Mathematics, University of Florence, Italy 1 [email protected] Department of Mathematics, University of Milan, Italy May 1, 2006
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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 Parity-check and decoding 3 Cyclic codes Definitions Algebraic structure Roots of unity and defining set An example Parity-check matrix BCH codes 4 Encoding and decoding with cyclic codes Polynomial encoding Decoding References
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An overview on error correcting codes Linear codes Cyclic codes Encoding and decoding with cyclic codes A communication schema B A SOURCE DESTINATION INFORMATION
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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
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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
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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
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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
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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 k -dimensional vector sub-space 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 | v , c = 0 c C } Is called the dual code of C .
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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 k -dimensional vector sub-space 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 .
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