By Volodia Blinovsky

Asymptotic Combinatorial Coding Theory is dedicated to the research of the combinatorial houses of transmission structures utilizing discrete signs. The ebook provides result of curiosity to experts in combinatorics trying to follow combinatorial how you can difficulties of combinatorial coding thought.
Asymptotic Combinatorial Coding Theory serves as an exceptional reference for resarchers in discrete arithmetic, combinatorics, and combinatorial coding idea, and should be used as a textual content for complex classes at the subject.

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Let us estimate (r A (L )) : (rA(L)) < [t, t s~+i(M(n) + s~+1-i(M(n) - - SJl)l+1-i SJl)l+iCit] [M(n)(M(n) - 1) ... (M(n) - L)t 1 n = (M(n))L+1[M(n)(M(n) - 1) ... 14 ) i=1 where ,\Jl = sJl/M(n). ;))Clt] , ,\ E [0,1], and n (M(n))L+1(M(n)(M(n) - 1)(M(n) - 2) ... 14 ) . ). ) coincide. )l-i) i=l 1-1 1-1 >. )i + >. )-i)C;t1' i=O This expression coincides with the expression in square brackets in the definition of '). Next statements we prove for fe(>'). 15 ) 3). (1 - >. ))l-l . 16 ) LIST DECODING 15 The lemma can be proved by induction on e (Problem 3).

13 ) is still valid if we sustitute parameter PA (L) for T A (L). Let M n = IAnl, where An C F n . 20 ) 16 ASYMPTOTIC COMBINATORIAL CODING THEORY and then to use the suggestions from the proof of Theorem 3. For arbitrary s points XI, ... ,X s , Xi = (xf, ... , xi) E F n define the scalar product ,(XI, ... , x s ) of sth order by the equality n ,(XI, ... ,X s )= Lx{ ... x{. j=1 We say that the code A is L-equidistant if for all s :S L scalar products of sth order depend only on s rather than on the choosing of vectors from the code.

AL+I) = 1- 112 - Lf~ll a~ I 2 L+ 1 . = From the definition of the ensemble A it follows that random variables On) = P (nE( on) nE(¢l) - t¢i) ) } exp( -(0 - E(

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