By Volodia Blinovsky

*Asymptotic Combinatorial Coding Theory* is dedicated to the research of the combinatorial houses of transmission structures utilizing discrete signs. The publication offers result of curiosity to experts in combinatorics looking to follow combinatorial easy methods to difficulties of combinatorial coding concept. *Asymptotic Combinatorial Coding Theory* serves as a very good reference for resarchers in discrete arithmetic, combinatorics, and combinatorial coding idea, and will be used as a textual content for complicated classes at the subject.

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79 ) + 1 > L. R(n+n,)FB(P, L). 80 ) = (xl, x[) EA. Let be the numbers of the messages Xii (yl) E A such that for all i j E Um +1 the following relations hold L y2:(y I ,y2)Eryi p(y2 I xl) ~ L y2:(y I ,y2)Ery, :f ij (yl ), where p(y2 I xli (yl )). 81 ) we obtain FA(p, L) ~ f{-l(n K(n+n') L + n') L p(yl I xI) i=l;i;tij(yl),jEU m +, y'EFn Pn,(p, log(m + l)ln', L) ~ Fn(p, R(l + n' In), m) Pn,(p, log(m + l)ln', L). X x Lemma 8 is proved. We take the logarithm of both sides of the last relations In Fn+n,(p, R, L) ~ In Fn(p, R(l + n' In), m) + In Pn,(p, log(m+ l)ln',L).

_1)m+i+ m- i=O m=L-i i ~ L.... 23 ) im .. L L(_1)m+i C:,,( No l=l+l m=1 L. 24 ) 'Y{li1 ... i m }). im> ... >i,>l The sum (No + N 1) is equal to d(X1' y). 24 ). 21 ) is proved in a similar way. 25 ) i=l where 'Y( i) denotes the scalar product of ith order and gi is defined as follows: i = 1, l*I, £+ 1 ~ i ~ 2£. To prove the corollary it is convenient to use the considerations given above taking into account that the scalar products of the same order are equal. )i). ))i i=l Problem 4 offers to prove that the last expression has the expansion L;~l giAi. *

Let also TJi be an indicator of the set Zi. The value of TJ = c~t~) L i=1 TJi on the code A is the number of different (L + 1)-subsets from A whose average radius is less than r. It follows from the definition of the ensemble A that the variables TJi have the same distribution. 4 ) Let us fix such a code A. Denote J ~ {i : A C Zi}. Let YeA be an arbitrary set of minimal cardinality such that 1Y Wi 1= 1 for all i E J. It is easy to see that A \ Y is the code whith minimum average radius not less than r.