By G. Cohen, I. Honkala, S. Litsyn, A. Lobstein
The issues of making masking codes and of estimating their parameters are the most situation of this ebook. It offers a unified account of the newest conception of masking codes and exhibits how a few mathematical and engineering concerns are regarding overlaying problems.Scientists enthusiastic about discrete arithmetic, combinatorics, laptop technological know-how, info idea, geometry, algebra or quantity conception will locate the booklet of specific importance. it's 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 study initiatives also are mentioned.
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Extra resources for Covering Codes
The problem is then to find a m i n i m u m - perfect - covering by packings. 10 S u b s e t s u m s a n d C a y l e y g r a p h s : Let a code have parity check matrix H with columns h i , . . , h,~, where each hi is a binary mtuple. As it will be shown, the covering radius of the code equals the minimal number r such that every binary m-tuple can be expressed as a componentwise sum modulo 2 of at most r columns of H. 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 + .
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. For example, if x - 1011000, then I-Ix T - (1, 1, 0) T and changing the sixth coordinate gives the codeword 1011010.
11; index the codewords with the 4-bit vectors, then replace each 7-bit vector with the index of the closest codeword, and afterwards retrieve the original vector with at most one error, since the H a m m i n g code has covering radius one. Thus, at most I/7 of the original long vector is distorted. In general, using a covering of the n-dimensional H a m m i n g space with 2 k spheres of radius r gives a compression ratio/c/n along with a distortion rate at most r/n. 2 D a t a c o m p r e s s i o n : Assume given a set of sixteen different 7-bit vectors - - their order being irrelevant m that we want to compress without distortion.