lect7 - Error Correcting Codes: Combinatorics, Algorithms...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 4

lect7 - Error Correcting Codes: Combinatorics, Algorithms...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online