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# lect2 - Error Correcting Codes Combinatorics Algorithms and...

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Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 2: Error Correction and Channel Noise August 29, 2007 Lecturer: Atri Rudra Scribe: Yang Wang & Atri Rudra As was mentioned in the last lecture, the fundamental tradeoff we are interested in for this course is the one between the amount of redundancy in the code vs. the number of errors that it can correct. We defined the notion of rate of a code to capture the amount of redundancy. However, before we embark on a systematic study of the tradeoff above, we need to formally define what it means to correct errors. We do so next. 1 Error correction Before we define what we mean by error correction, we formally define the notion of encoding . Definition 1.1 (Encoding function). Let . An equivalent description of the code is by an injective mapping called encoding function. Next we move to error correction. Intuitively, we can correct a received word if we can recover the transmitted codeword (or equivalently the corresponding message). This “reverse” process is achieved by decoding . Definition 1.2 (Decoding function). Let be a code. A mapping is called a decoding function for . The definition of a decoding function by itself does not give anything interesting. What we really need from a decoding function is that it recovers the transmitted message. This notion is captured next. Definition 1.3 (Error Correction). Let and let be an integer. is said to be - error-correcting if there exists a decoding function such that for every error message and error pattern with at most errors, . Figure 1 illustrates how the definitions we have examined so far interact. We will also very briefly look at a weaker form of error recovery called error detection .

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