By Jørn Justesen and Tom Høholdt

This publication is written as a textual content for a direction aimed toward complicated undergraduates. just some familiarity with uncomplicated linear algebra and chance is without delay assumed, yet a few adulthood is needed. the scholars might concentrate on discrete arithmetic, desktop technological know-how, or verbal exchange engineering. The publication is additionally an appropriate creation to coding conception for researchers from similar fields or for pros who are looking to complement their theoretical foundation. It provides the coding fundamentals for engaged on initiatives in any of the above parts, yet fabric particular to at least one of those fields has now not been integrated. Chapters conceal the codes and deciphering equipment which are at present of so much curiosity in learn, improvement, and alertness. they offer a comparatively short presentation of the fundamental effects, emphasizing the interrelations among various equipment and proofs of all very important effects. a chain of difficulties on the finish of every bankruptcy serves to study the implications and provides the scholar an appreciation of the suggestions. moreover, a few difficulties and proposals for initiatives point out path for additional paintings. The presentation encourages using programming instruments for learning codes, enforcing deciphering equipment, and simulating functionality. particular examples of programming workout are supplied at the book's domestic web page. allotted in the Americas by way of the yank Mathematical Society.

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**Additional resources for A Course in Error-Correcting Codes (EMS Textbooks in Mathematics)**

**Sample text**

That the nonzero elements of S have a multiplicative inverse (axiom 8) can be seen in the following way: Let x be a nonzero element of S and consider the set Sˆ = {1 · x, 2 · x, . . , ( p − 1) · x}. We will prove that the elements in Sˆ are different and nonzero so therefore Sˆ = S\{0} and in particular there exists an i such that i · x = 1. It is clear that 0 ∈ / Sˆ since 0 = i · x, 1 ≤ i ≤ p − 1 implies p|i x and since p is a prime therefore p|i or p|x, a contradiction. To see that the elements in Sˆ are different suppose that i · x = j · x where 1 ≤ i, j ≤ p − 1.

1. Reed-Solomon codes over F11 Since 2 is a primitive element of F11 we can take xi = 2i−1 mod 11, i = 1, 2, . . 2 Decoding Reed-Solomon Codes 51 So (1, 0, 0, 0, 0) (0, 1, 0, 0, 0) (0, 0, 1, 0, 0) (0, 0, 0, 1, 0) (0, 0, 0, 0, 1) is encoded into is encoded into is encoded into is encoded into is encoded into (1, 1, 1, 1, 1, 1, 1, 1, 1, 1) (1, 2, 4, 8, 5, 10, 9, 7, 3, 6) (1, 4, 5, 9, 3, 1, 4, 5, 9, 3) (1, 8, 9, 6, 4, 10, 3, 2, 5, 7) (1, 5, 3, 4, 9, 1, 5, 3, 4, 9) and these five codewords can be used as the rows of a generator matrix of the code, which is a (10, 5, 6) code over F11 .

7). 38 Bounds on error probability for error-correcting codes Because most errors occur for j close to w2 , we can use this value of j for all error patterns. 9) w>0 This is a useful bound when the number of errors corrected is fairly large. 9) indicate that the error probability depends not only on the minimum distance, but also on the number of low weight codewords. As p increases, codewords of higher weight may contribute more to the error probability, because their number is much greater. 7) we overestimate the error probability whenever a vector is closer to more than one nonzero word than to the zero word.