Download Covering Codes by G. Cohen, I. Honkala, S. Litsyn, A. Lobstein PDF

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).

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