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

The issues of creating protecting codes and of estimating their parameters are the most hindrance of this publication. It offers a unified account of the newest concept of overlaying codes and indicates how a few mathematical and engineering concerns are on the topic of protecting problems.

Scientists occupied with discrete arithmetic, combinatorics, laptop technological know-how, details conception, geometry, algebra or quantity idea will locate the e-book of specific importance. it's designed either as an introductory textbook for the newbie and as a reference publication for the specialist mathematician and engineer.

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

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**Example text**

For a code C with minimum distance d, the integer e- [(d--1)/2] is called the packing radius or the error-correcting capability of the code C. Notice that e is the largest integer such that the Hamming spheres of radius e centred at the codewords are disjoint. Therefore IcI _< q'~lVq(n, e), which is called the sphere-packing bound or Hamming bound. If C has packing radius e and covering radius R, then clearly e _~ R. Moreover, if d is even, then e < R, and therefore we always have d<2R+l. A code (7 C_ Q'~ with K codewords, minimum distance d and covering radius R is called an (n, K, d)qR code.

By linearity, the projection of C on any vector x E IFTM, m ~ n, contains only even vectors, or the same number of even and odd vectors. Moreover, Px(C) consists of only even vectors if and only if x E C • Thus, r _ f ICl i~EC', 0 ifx ~ C • Chapter 2. 6) : c 6 C x6Sj The inner sum in the last expression does not depend on the particular choice of e, but only on the weight w(e). If w(e) - i, then each of the (~) (j-t)'~-i vectors of weight j with exactly ~ l's in common with e contributes ( - 1 ) l to the inner sum.

0). A vector with even (odd) weight is caned even (odd). The set of even (odd) codewords of a code C is called the even weight (odd weight) subcode of C and is denoted by Ce (Co). The support of a vector x E Q'~ is the set {i" xi r 0} and is denoted b y s u p p ( x ) . In the case of a binary alphabet, vectors can be identified with their supports, and for two binary vectors, x and y, of the same length, x C y, x u y, x A y and x \ y refer to the supports of x and y. , d - d(C) - min a,bEC,a~b d(a, b).