lect4 - Error Correcting Codes: Combinatorics, Algorithms...

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Unformatted text preview: Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 4: Hamming code and Hamming bound September 5,2007 Lecturer: Atri Rudra Scribe: Kanke Gao & Atri Rudra In the last couple of lectures, we have seen that the repetition code C 3 ,rep , which has distance d = 3 , can correct ≤ 1 error. On the other hand the parity code C ⊕ , which has distance d = 2 , can detect ≤ 1 error, but can not correct 1 error. In the last lecture, we extended these observations to the general case: code with distance d can correct ⌊ d − 1 2 ⌋ errors and can detect d − 1 errors. Thus, the fundamental tradeoff that we are interested in (the amount of redundancy in the code vs. the number of error that it can correct) is equivalent to the one between rate and distance of the code (for worst-case errors). 1 Hamming Code We have seen that the repetition code C 3 ,rep has distance 3 and rate 1 / 3 . A natural question to ask is whether we can have distance 3 with a larger rate. With this motivation, we will now consider the so called Hamming code (named after its inventor, Richard Hamming), which we will denote by C H . Given a message ( x 1 , x 2 , x 3 , x 4 ) ∈ { , 1 } 4 , its corresponding codeword is given by C H ( x 1 , x 2 , x 3 , x 4 ) = ( x 1 , x 2 , x 3 , x 4 , x 2 ⊕ x 3 ⊕ x 4 , x 1 ⊕ x 3 ⊕ x 4 , x 1 ⊕ x 2 ⊕ x 4 ) , where the ⊕ denotes the EXOR operator. It is easy to check that this code has the following parameters: C H : q = 2 , k = 4 , n = 7 , R = 4 / 7 . Before we move onto determining the distance of C H , we will need another definition....
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This note was uploaded on 10/16/2011 for the course ELECTRICAL EE251202 taught by Professor Rejaei during the Spring '10 term at Sharif University of Technology.

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lect4 - Error Correcting Codes: Combinatorics, Algorithms...

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