By A. Bak

ISBN-10: 0387128913

ISBN-13: 9780387128917

ISBN-10: 3540128913

ISBN-13: 9783540128915

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**Extra info for Algebraic K-Theory, Number Theory, Geometry, and Analysis: Proceedings**

**Sample text**

1. 1). Due to our time limitation we have a serious problem! There is no point in sending each symbol twice instead of once. A most remarkable theorem, due to C. E. Shannon (cf. [62]), states that, in the situation described here, we can still achieve arbitrarily small error probability at the receiver. The proof will be given in the next section. A first idea about the method of proof can be obtained in the following way. We transmit the result of two tosses of the coin as follows: heads, heads --+ 0 0 0 0, heads, tails --+ 0 1 1 1, tails, heads --+ 1 0 0 1, tails, tails --+ 1 1 1 O.

Majority Logic Decoding In this section we shall briefly sketch a decoding method which is used with many linear codes. Generalizations will occur in later chapters. The method is simple and it has the advantage that in some cases more errors are corrected than one expects. 1) Definition. ) "# 0 for at most one value of v. Now suppose x is a received word that contains t errors, where t ~ Then "or {~ t values of v, if Xi is correct, (x, y(V» "# 0 11 ~ r - (t - 1) values of v, ifxi is incorrect.

The author gratefully acknowledges that he acquired a large part of his knowledge of coding theory during his many visits to Bell Laboratories. The reader interested in more details about the code used in the Mariner 1969 is referred to reference [56]. For the coding in Compact Disc see [77], [78]. By consulting the references the reader can see that for many years now the most important results on coding theory have been published in IEEE Transactions on Iriformation Theory. 4. 1. 1). 2. 2. We showed that the probability that a received word is decoded correctly is q6 + 6qSp + q4 p2.

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