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 making masking codes and of estimating their parameters are the most quandary of this ebook. It offers a unified account of the newest conception of protecting codes and exhibits how a couple of mathematical and engineering matters are with regards to protecting problems.Scientists concerned about discrete arithmetic, combinatorics, laptop technological know-how, details concept, geometry, algebra or quantity conception will locate the publication of specific value. it truly is 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 compatible for study initiatives also are mentioned.

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XnYn). We t h e n have the obvious f o r m u l a d(x, y) In the b i n a r y case x + y - x - y - w(x- y). 5) + y). 6) and y) - For two sets A, B C_ 7/qn or A, B C_ ]F nq we d e n o t e A+B-{a+b'aCA, TI, b C B}. n For x C 7/q or x E IF q the set x + A- { x + a ' a E A} is called a translate of A. Clearly, a code a n d its t r a n s l a t e have the s a m e covering radius. More generally, two equivalent codes have the s a m e covering radius. In IFq we can also define a scalar multiplication: if x - (Xl, x 2 , .

Therefore K ( R , R ) - 1 and g ( n , R ) ~_ 2 when n > R. 10 has covering radius Ln/2J <_ R. 15 (i) If there is a binary [n, k]R code and R ~ R' ~ n, then there also exists a binary In, kt]R ~ code with k t ~ k. (ii) If, furthermore, R' <_ n - k, then there exists a binary [n, k]R' code. P r o o f . (i) Change l's to O's in the k • n generator matrix, one at a time. When there are only O's left, the matrix generates a code with covering radius n. Each one-bit change in the generator matrix alters at most one bit in each codeword, and therefore at each step the covering radius changes by at most one.

1. 5. T h e complement ~ of a b i n a r y vector x E IF '~ is o b t a i n e d by c h a n g i n g all the zeros to ones a n d vice versa. In the s a m e way as 0 - 0 '~ - (0, 0 , . . , 0) we d e n o t e 1 - 1~ - (1, 1 , . . , 1). More generally, 0 i l j, for instance, d e n o t e s a vector b e g i n n i n g with i zeros a n d ending with j ones. We also use the notations el lO s - l , e2 -- 010 n - 2 , . . , e n - - 0 n - l l . If our a l p h a b e t is 7/q or IFq, we can define the s u m a n d difference of two vectors x - (Xl, x 2 , .

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