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 , .