Number theory for computing : with 33 tables, Edition: 2. by Song Y Yan; Martin E Hellman

By Song Y Yan; Martin E Hellman

Foreword via Martin E. Hellman.- Preface to the second one Edition.- Preface to the 1st Edition.- 1. effortless quantity Theory.- 2. Computational/Algorithmic quantity Theory.- three. utilized quantity Theory.- Bibliography.- Index

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Source I I Encoder Decoder Receiver j Codcword\ /Codeword X1X2 \ Noisy channel Fig. / Y, 1 The source will produce a message consisting of a sequence of source symbols, and this message is to be transmitted to its intended receiver across a noisy channel. Without any real loss of generality, we assume that the channel has the same alphabet I, of size q, for input and output. A code over I is a collection of sequences of symbols from I; the members of are codewords. We assume that all codewords are of the same length.

D'1 — n1D'2 — . — n,_2D, S D'2 — n1D'3 —... 1 D3 — n1D2 D2 n1 SD. 3 inequalities are the key to constructing a code with the given word lengths. We first choose n1 words of length 1, using distinct letters from I. This leaves D — n1 symbols unused, and we can form (D — n1)D words of length 2 by adding a letter to each of these. These Choose our n2 words of length 2 arbitrarily from these, and this — n2 prefixes of length 2. These can be used to form (D2 — n1D — n2)D words of length 3, from which we choose n3 arbitrarily, and so on.

2 1. A message consisting of N binary digits is transmitted through a binary symmetric channel having error probability p. Show that the expected number of errors is Np. Connecting the source to the channel Consider the following situation: we have a memoryless source 97 which emits symbols (or source words) s1,. , SN with probabilities source is connected to a binary symmetric channel This PN. Pi, . with error probability p as shown: q [Decoder We assume that the encoding into binary is noiseless, and is known to the decoder.

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