By G. Cohen, I. Honkala, S. Litsyn, A. Lobstein
The issues of creating protecting codes and of estimating their parameters are the most situation of this ebook. It presents a unified account of the newest concept of masking codes and exhibits how a few mathematical and engineering matters are on the topic of protecting problems.
Scientists concerned with discrete arithmetic, combinatorics, desktop technology, info conception, geometry, algebra or quantity concept will locate the publication of specific importance. it really is designed either as an introductory textbook for the newbie and as a reference booklet for the specialist mathematician and engineer.
A variety of unsolved difficulties compatible for learn initiatives also are mentioned.
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Extra resources for Covering Codes
Sample text
It is a particular case of the following group-theoretic problem: Given an abelian group (G, +), and a generating subset S, containing zero, find the minimal r (covering number of G) such that S + S + . . + S - G, where the sum extends over r copies of S. The set (hi} can be used for constructing the Cayley graph P as follows. Let the 2TM vertices of the graph be associated to all binary m-tuples, with an edge between two vertices if and only if their componentwise sum modulo 2 is a column of H.
Consider how the 16 vectors in the H a m m i n g space beginning with 0 are covered by the codewords. Notice t h a t a codeword beginning with 0 covers exactly five of them, namely the codeword itself and the four vectors obtained by changing one of the last four coordinates. A codeword t h a t begins with 1 only covers one such vector, the one obtained by changing the first coordinate from 1 to 0. But this means t h a t all the codewords together can cover at most 5 . 5 ~ 6 . 1 - 31 < 2- 16, a contradiction.
10 The binary In, n - H-( 1 1] code C with parity check matrix 1 ... 1) consists of all the even words of length n. It is called the even weight code and is denoted by IE'~. Its covering radius is equal to 1. The dual code C a- Chapter 2. Basic facts 22 generated by H is called the repetition code. It has parity check matrix 1 1 it(c • . - 1 1 @ 9 @ 1 1 1 1 which is also a generator matrix of C. From the definition it is clear that the covering radius of C • which only consists of the words 0 and 1, is equal to [n/2J.