By G. Cohen, I. Honkala, S. Litsyn, A. Lobstein

The issues of making overlaying codes and of estimating their parameters are the most problem of this booklet. It offers a unified account of the latest conception of protecting codes and exhibits how a few mathematical and engineering matters are relating to protecting problems.

Scientists enthusiastic about discrete arithmetic, combinatorics, desktop technology, info concept, geometry, algebra or quantity concept will locate the e-book of specific value. it's designed either as an introductory textbook for the newbie and as a reference booklet for the professional mathematician and engineer.

A variety of unsolved difficulties compatible for study initiatives also are mentioned.

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**Sample text**

11 The following Hamming code of length seven and size 16 is truly exceptional: it turns out t h a t the spheres of radius one centred at the codewords not only cover the whole space, but are also pairwise disjoint. Such a code is called perfect. It is linear, with parity check m a t r i x tI- 0 1 1 0 1 1 0 0 0 1 1 0 1 1 . Notice t h a t all the nonzero vectors of length three appear as columns in I-I. Let x be any vector of length seven. If I-Ix T = 0, then x itself is a codeword. Otherwise, I'Ix T is one of the columns of I-I, and changing the corresponding coordinate in x yields a codeword at distance one to x.

But then the result follows by applying the first case a suitable number of times. [] An easy modification of the previous proof yields the corresponding statement about the union of three Hamming spheres. 11 Assume that r, s, t E {0, 1 , . . , n), el, c2, c3, c~, c~, c~ C r <_ u rt aZl u j. <_ U B, Chapter 2. Basic facts 36 P r o o f . We proceed in the same way as in the previous proof. If in the first case x ~ B~(Cl)U Bs(e2) belongs to Bt(c3) \ Bt(c~), then xl = x2 = 0 and changing the first two coordinates in x gives a vector x' ~ B ~ ( c l ) U B8(c2) that belongs to Bt(c~) \ Bt(c3).

For example, we can take 7/q - the set of integers modulo q , endowed with the usual addition and multiplication. If p is a prime, every element in 7/p has a multiplicative inverse, and so 7/v is a field, which we also denote by IFp. Informally, a field is a set where addition, subtraction, multiplication and division by a nonzero element are defined and satisfy the usual rules. More generally, a finite field of q elements exists whenever q is a prime power, and for each prime power q such a field is essentially unique.