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# cs - Introduction to Algebraic Coding Theory Supplementary...

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Introduction to Algebraic Coding Theory Supplementary material for Math 336 Cornell University Sarah A. Spence Contents 1 Introduction 1 2 Basics 2 2.1 Important code parameters . . . . . . . . . . . . . . . . . . . . . 4 2.2 Correcting and detecting errors . . . . . . . . . . . . . . . . . . . 5 2.3 Sphere-packing bound . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Linear codes 9 3.1 Generator and parity check matrices . . . . . . . . . . . . . . . . 11 3.2 Coset and syndrome decoding . . . . . . . . . . . . . . . . . . . . 14 3.3 Hamming codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Ideals and cyclic codes 19 4.1 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Cyclic codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 Group of a code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.4 Minimal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.5 BCH and Reed-Solomon codes . . . . . . . . . . . . . . . . . . . 33 4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5 Acknowledgements 40 1 Introduction Imagine that you are using an infrared link to beam messages consisting of 0s and 1s from your laptop to your friend’s PalmPilot. Usually, when you send a 0, your friend’s PalmPilot receives a 0. Occasionally, however, noise on the channel causes your 0 to be received as a 1. Examples of possible causes of noise include atmospheric disturbances. You would like to find a way to transmit your messages in such a way that errors are detected and corrected. This is where error-control codes come into play. 1

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Error-control codes are used to detect and correct errors that occur when data is transmitted across some noisy channel. Compact discs (CDs) use error- control codes so that a CD player can read data from a CD even if it has been corrupted by noise in the form of imperfections on the CD. When photographs are transmitted to Earth from deep space, error-control codes are used to guard against the noise caused by lightning and other atmospheric interruptions. Error-control codes build redundancy into a message. For example, if your message is x = 0, you might encode x as the codeword c = 00000. (We work more with this example in Chapter 2.) In general, if a message has length k , the encoded message, i.e. codeword, will have length n > k . Algebraic coding theory is an area of discrete applied mathematics that is concerned (in part) with developing error-control codes and encoding/decoding procedures. Many areas of mathematics are used in coding theory, and we focus on the interplay between algebra and coding theory. The topics in this packet were chosen for their importance to developing the major concepts of coding theory and also for their relevance to a course in abstract algebra. We aimed to explain coding theory concepts in a way that builds on the algebra learned in Math 336. We recommend looking at any of the books in the bibliography for a more detailed treatment of coding theory. As you read this packet, you will notice questions interspersed with the text. These questions are meant to be straight-forward checks of your understanding. You should work out each of these problems as you read the packet. More homework problems are found at the end of each chapter.
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