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Unformatted text preview: Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 7: Family of Codes Sep 12, 2007 Lecturer: Atri Rudra Scribe: Yang Wang & Atri Rudra In the previous lecture, we were going to see which codes are perfect codes. Interestingly, the only perfect codes are the following: • The Hamming codes which we studied in the last couple of lectures, • The trivial [ n, 1 ,n ] 2 codes for odd n (which have n and 1 n as the only codewords), • Two codes due to Golay [1]. The above result was proved by van Lint [3] and Tietavainen [2]. In today’s lecture, we will look at an efficient decoding algorithm for the Hamming code and look at some new codes that are related to the Hamming codes. 1 Family of codes Till now, we have mostly studied specific codes, that is, codes with fixed block lengths and dimen sion. The only exception was the “family” of [2 r 1 , 2 r r 1 , 3] 2 Hamming codes (for r ≥ 2 ). The notion of family of codes is defined as following: Definition 1.1 (Family of codes) . C = { C i } i ≥ 1 is a family of codes where C i is a [ n i ,k i ,b i ] q code for each i (and we assume n i +1 > n i ). The rate of C is defined as R ( C ) = lim i →∞ k i n i . The relative distance of C is defined as δ ( C ) = lim i →∞ d i n i . For example, C H the family of Hamming code is a family of codes with n i = 2 i 1 ,k i = 2 i i 1 ,d i = 3 and R ( C H ) = 1 ,δ ( C H ) = 0 . We will mostly work with family of codes from now on. This is necessary as we will study the asymptotic behavior of algorithms for codes, which does not make sense for a fixed code. For example, when we say we say that a decoding algorithm for a code C takes O ( n 2 ) time, we would be implicitly assuming that C is a family of codes and that the algorithm has an O ( n 2 ) running time when the block length is large enough. From now on, unless mentioned otherwise, whenever we talk about a code, we will be implicitly assuming that we are talking about a family of codes. Finally, note that the motivating question is to study the optimal tradeoff between R and δ ....
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 Spring '10
 Rejaei
 Electromagnet

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