On the extremal combinatorics of the hamming space

JOURNALOF COMBINATORIALTHEORY,Series A 71, 112-126 (1995)

On the Extremal Combinatorics of the Hamming Space JANos K 6 R ~ R * Dipartimento di Mathematica, Bologna University, Bologna, Italy Communicated by the Managing Editors Received November 3, 1993

We present new asymptotic bounds for problems in extremal set theory related to finding the maximum number of qualitatively 3-independent bipartitions of an n-set. We consider the space of all binary sequences of some fixed length n. As we select subsets of growing cardinality we see an increasing number of different "small configurations" necessarily emerge. © 1995AcademicPress, Inc.

1. ABOUT LANGUAGE--AN APOLOGY Combinatorics is cute. It speaks a b o u t graphs and hypergraphs, things y o u can draw and see. However, in mathematics it is often necessary or, to say the least, useful to c o m p a r e questions in order to reduce the u n k n o w n to what is already known, to find the natural place for a problem. In fact, once we have correctly identified the right context for our particular problem, the bulk of knowledge in that context starts to suggest ideas for the solutions. We will argue that the natural context for the problems we intend to treat in this paper is neither graphs nor hypergraphs, but sequences of fixed length over a finite alphabet. As we shall treat well-known problems concerning subsets of a set or vertices of the hypercube or even just simple graphs, we will c o m p a r e the questions and get new answers by switching from one p r o b l e m to the other. In order to do this, we will consider subsets or bipartitions of an n-set, or the vertices of the n-dimensional hypercube, as well as the edges of a graph on n vertices, merely as binary sequences of length n. Some of our problems will be cute. As y o u can guess, these are the k n o w n ones. Here we will get some new answers. But we will find also new questions along the way; questions that have no nice short formulation in any of the "natural" languages of graphs, partitions, and the like. These will remain questions a b o u t sequences. 1 Present address: Dipartimento di Scienze dell'Informazione, Universifft "La Sapienza", via Salaria 113, 00198 Rome, Italy. 112 0097-3165/95 $12.00 Copyright © 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.

COMBINATORICS OF THE HAMMING SPACE

1 13

Some problems are more natural than others even in a literal sense. They keep coming back in various contexts just because they are out there in some objective way, without our making them up. We present these first, starting for each of them with its most attractive formulation, followed by the rest.

2.

GEOMETRY OF THE H A M M I N G SPACE

The set of n-length binary sequences can be endowed with the Hamming metric. In this metric, the distance between two sequences is the number of coordinates in which they differ. Once you have a metric, you can define spheres. A very attractive result of Katona [24] (in fact, a combination of an old result of Harper [ 19] and the celebrated Kruskal-Katona theorem [36, 22]) says that in the Hamming space, just as in the more familiar Euclidean space, the sphere has the isopermetric property of yielding the minimum "surface" among the sets of given "volume." By now, there is a steady flow of papers on various generalizations of the isopermetric property within combinatorics, but this is not our topic here. Rather, we introduce a new problem in this geometric setting. We say that three points in the Hamming space are on the same line if they satisfy the triangular inequality with equality. If this is not the case, we say that the three points are in a general position. PROBLEM A (Point Sets in General position). How many different points can we find in the n-dimensional Hamming space so that no three of them are on a line? Let us call their number A(n). As we shall soon see, this problem has been known for a long time in a different form, even though no answer is in sight. Before discussing this and presenting some new results, let us briefly mention that ours is not the only possible definition of a line in a combinatorial setting. (An analogous problem for a different definition of a (combinatorial) line leads to the beautiful Theorem E of F/irstenberg and Katznelson [ 13] which yields a far-reaching generalization of Szemer6di's fundamental theorem on arithmetic progressions in "dense" sets of integers [53].) PROBLEM B* (3 Partitions per Triple). What is the maximum number of points in a set X such that n bipartitions of X suffice to cut every triple of them into two (non-empty) parts in every possible way ? Let B(n) denote the maximum cardinality of such a set 35. Clearly, this problem is equivalent to a long-standing question of Friedman et al. [12]:

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JANOS KORNER

PROBLEM B ((1, 2)-Separation). Let X be a set and let N~ .... , N,, be a family of bipartitions of X. We say that the { ~}"i= form a (1, 2)separating system for X, if for every x e X , every {y, z} ~ (x) disjoint from x there is an i, 1 ~
LEMMA 1.

A(n) = B*(n) = B(n). Proof B*(n) = B(n) is trivial. To prove that A ( n ) = B(n) we represent an arbitrary family N1, ~2 .... , ~ , of bipartitions of X by a set of binary sequences in the obvious way. Let fi: X ~ { 0, 1 } generate the bipartition As, i.e., f i ( x ) = 0 if x e X belongs to whatever we would like to consider the "first" class of N~, and f i ( x ) = 1 else. Now gi = 1 - f i is another representation of ~ , and in fact it will not matter whether we choose f,. or g , Let {f,.} be the set of binary-valued functions so obtained. We define ]XI binary sequences of length n by associating with every element x e X the binary sequence

f(x)

=fa(x), f2(x) ..... L(x).

Reversing this correspondence, for any set X of binary sequences of length n we obtain n bipartitions of the set X. Consider the matching between the set of partitions and the set of sequences so obtained. We claim that the binary sequences are points in general position in the n-dimensional Hamming space if and only if they generate n bipartitions that form a (1, 2)-separating system for X. To see this, consider three different binary sequences, y, z, and w of length n. Take all the bipartitions of the 3-element set J ( = {y, z, w} generated in the coordinates of our three sequences. If only two of the three possible (non-trivial) partitions are present, this means that two sequences, say w.l.o.g, y and z, are never together in a class of a partition separating them from w. In other words, y~ = zi implies wi= yi for every coordinate i = 1, 2, ..., n. On the other hand, the latter means that

l Y i - Zil = [ y , - w i l + Iz~- w,I, and hence dH(y, z) = da(y, w) + dH(z, w),

(1)

C O M B I N A T O R I C S OF T H E H A M M I N G SPACE

115

where di~ stands for the Hamming distance. Thus the last equation expresses that the three points are on the same line. Reversing the above resoning we also see that if three points y, z, and w of the Hamming space are on the same line, with w being between y and z in the sense of relation (1), then the biparition separating {y, z} from w is not represented among the coordinates of these three sequences. | It is worth mentioning that Problem A can be formulated in the language of extremal set theory in the folowing way: PROBLEM A*. What is the maximum cardinality of a family ~ of subsets of the n-set In] = { 1, 2 ..... n} having the property that no three different elements A, B, C of Y satisfy B~C~_A~_BuC. Clearly, this is just a reformulation of Problem A. It does not seem to have been stated elsewhere. The determination of A(n) is a very hard combinatorial question. In fact, not even the exponential asymptotics 0~= lim sup -1 log A(n) n~o~

n

has been established. We shall give new bounds on ~, improving on the earlier results of Friedman and co-workers [11, 12]. (For the related problem of (2, 2)-separation and its relation to other problems in combinatorics we refer the reader to K6rner and Simonyi [ 33 ], where the very powerful linear programming bound for the code distance problem, obtained by McEliece et al. [38], is used in order to get a non-existence bound. We remark, however, that for excluded ternary configurations, such as ours here, the code distance bound cannot be applied in the same way.

3. EXTREMAL SET THEORY The following is a well-known problem in extremal set theory, introduced by Erd6s et al. [7] (cf. also the more general version [8] and the recent proof of Ruszink6 [47]). PROBLEM C (2-Cover-Free Families). A family ~- of subsets of the n-set [n] = { 1, 2 .... , n} is called 2-cover-free if for any three different sets A, B, C from the family we never have A~BwC. 582a/71/1-9

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JANOS KORNER

What is the maximum cardinality

C(n) of such a family ~-?

This problem is also treated independently by Hwang and S6s [20] and by Dyachkov and Rykov [ 6 ]. The last authors introduce it with a motiviation from multi-access communication as a problem of access allocation under the hypothesis that at most three senders can enter in a collision at any given time. For the details of this background, the reader is referred to [ 6]. The same problem is also known under the name of superimposed codes and goes back to Kautz and Singleton [ 26 ]. Recently, the results on this problem have been applied in locality-based graph coloring by Linial [ 37] and Szegedy and Vishwanathan [ 52]. Actually, the first paper on the topic is [55]. In the paper of Dyachkov and Rykov [6] the formulation of Problem C is the following: PROBLEM C* (Superimposed Codes).

What is the maximum number

C*(n) of binary sequences of length n that we can select with the property that for any three of them, y, z and w, say, there is a coordinate i such that Yi = 1, wi = zi = 0. Clearly, this is the same as Problem C, and we immediately have

C*(n) C(n). =

Furthermore, we have LEMMA 2.

Proof The second inequality is trivial. To prove the first one, note simply that if the partitions ~1, ~2 ..... ~z, l = kn/2_], form a (1, 2)-separating system for X, then, by completing the sequ.ences fl(x), fz(x), ..., fz(x) by setting fz+i(x)=l-fi(x), i= 1,2, ..., l, we obtain the characteristic vectors of a 2-cover-free family. The cardinality of this family is A(Ln/2J). If n is even, the n th coordinate of each sequence can be defined in an arbitrary manner. Once again, the exponential asymptotics 1

7 = lim sup - log n~oo

is unknown.

n

C(n)

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117

4. QUALITATIVE 3-1NDEPENDENCE

Rbnyi [45] posed (in a more general form) the following. PROBLEM D (Qualitatively 3-Indepenent Bipartitions). What is the maximum number D(n) of bipartitions of a set of n elements, if for any three of them the intersection of any three of their respective classes is non-empty ? If three bipartitions have the above property, they are said to be qualitatively 3-independent. This name refers to the fact that 3-independence is a sufficient and necessary condition for 3 bipartitions to give rise to 3 (totally) independent binary-valued random variables. The first bounds on D(n) are those of Kleitman and Spencer [27]. Recently, there has been a surge of interest in this problem, due to its relevance to the construction of small probability spaces, that is, a way to reduce the use of the resource "randomness" in algorithms; cf. the papers of Alon [ 1 ], Naor and Naor [39], Seroussi and Bshouty [48], Alon et al. [2], and Sloane [50]. This problem is just as hard as all the previous ones, and the asymptotics of D(n), i.e., 1 d = lim sup - log D(n), n~o~

n

is still unknown. (It is all the more remarkable that the general problem of finding the asymptotics of 2-independent k-partitions of an n-set can be determined and the result is a strikingly simple formula, cf. Gargano et al. [15], where references to the earlier work of the Prague school of combinatorics [41-44] can be found. The result in [15] is based on an information-theoretic construction technique whose power is best explained with far more generality in Gargano et al. [ 16].) In order to insert R6nyi's problem into our framework, we introduce the equivalent PROBLEMD* (Surjectivity). What is the maximum number D*(n) of binary sequences of length n with the property that for any three of them, y, z, and w, and any sequence a e {0, 1 } 3 there is coordinate such that YiZiWi

~ a.

The equivalence of the two preceding problems should be clear, and thus also D(n)=D*(n). This formulation is the one used in the context of switching circuits, cf. Sloane [50], and has a considerable literature to which some references are contained in Seroussi and Bshouty [48] and Sloane [50].

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A third formulation of this problem is geometric and comes from computer science. In recent years there has been growing interest designing fault-tolerant computer architectures for massively parallel computations, cf. e.g. Becker and Simon [ 3 ] and Graham et al. [ 18 3. The latter contains a partial update on the literature. PROBLEM E (Hypercube Architecture). What is the maximum dimension E(n) of a hypercube such that if we remove n - 1 of its vertices in whichever way we want, the residual structure will still contain a sub-hypercube of dimension E(n) - 3 ? It is well-known and easy to see (cf. e.g. [ 18]) that E(n)= D(n). We are concerned here with another relationship, that of Problems A and D LEMMA 3.

Proof The only non-trivial (although very simple) inequality is the first one. As in the proof of the previous lemma, let the partitions N1,..., NI, l= [_n/2_]- 1, form a (1, 2)-separating system for X, Ixl = l. Complete the corresponding sequences fl(x), ...,fl(x), x e X, by adding fz+i = 1 - f i ( x ) , i= 1, 2, ..., l, f2t+l(X) = 0,f2z+z(X) = 1. (Ifn is odd, proceed as in Lemma 2.) The rows of the resulting array, i.e., the entries corresponding to the same x, will form a set of sequences in {0, 1} ~ that satisfies the conditions of Problem D*. | The preceding discussion has hopefully convinced the reader of the relevance of Problem A to all of the other problems treated so far.

5. BOUNDS Strangely enough, the problem of determining A(n) or even c~ has received far less attention than those problems concerning 7 and & (We believe that to some extent this is due to the lack of a "good story" associated with the problem). Before introducing our new result, let us quote the known asymptotic bounds. Fredman and Koml6s [ 11 ] (unlike Friedman et al. [12]) do not pay any particular attention to the case of (1, 2)-separation. On the other hand, the known results on ~ will yield a non-trivial upper bound on 0¢, to be improved in the following. Erdrs et al. [ 7 ] proved that 0, 181... ~<), ~
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119

The same upper bound was also established in the earlier paper [6] by D y a c h k o v and Rykov. The lower bound was derived using r a n d o m selection from all the n-length sequences with a fixed number kn of l's for a suitable k. The main focus has been, however, on finding the exact value of d. Here the best known bounds read 0.132... ~<,~~
....

(3)

The upper bound is due to Kleitman and Spencer [34], while the lower bound is hard to attribute. It appears in the Sloane paper with a reference to unpublished work of Roux [46]. It is also contained in G r a h a m et al. [ 18], traced back to Erd6s et al. [7]. In fact, G r a h a m et al. just fill in the missing numerical calculations in [7]. Finally, by L e m m a 3, the upper bound of (3) implies ~<2(h(1/4) - 1) = 0.622 ....

(4)

By a direct approach, we will get a better upper bound on ~. The idea of its proof is related to that of the analogous result in [ 7]. We will also derive a lower bound o n , by random choice. Our bounds read THEOREM 1.

0.21... ~c~< ~1 Proof

Let us prove first that .
~ 5.

(5)

To this end, fix some n, and consider a set ~_~ { 0, 1 } n of points in general position whose cardinality is m a x i m u m with this property. Dividing ~ into sets CI, such that x e Ck if ~7= a = k, we see that for some k

I~1

IGI ~ n + l "

(6)

Given any x e Q , we consider all the ternary sequences y e { 0 , 1,2} n such that, roughly speaking, half of the coordinates in which xi = 0 as well as half of the coordinates in which x i = 1 are changed into 2's, while the rest of them remain unchanged. More precisely, l of the l's and m of the O's, say are changed, where l and m are to be chosen later. If some x and y are in this relationship, then we will say that y covers x. The number of pairs (x, y) such that x ~ C~, and y covers x is k

n-k

m I.

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J~NOS KORNeR

Now, each q-ary sequence generates a q-partition of the set [n] in an obvious manner. We denote by ~ the partition generated by the sequence y. Further, we denote by f ( x , y) the 4-partition of [n] generated as the common refinement of the partitions ~ and ~. Clearly, the 4-partition f ( x , y) does not necessarily determine either the sequence y, or the 3-partition ~. Still, for any given x ~ there can be at most 4 different sequences y producing the same pair (x, f ( x , y)). We call a sequence y a signature, if for some x ~ and any (x*, y*) the equality y * = y implies x* = x. We claim that at least one of the at most four different sequences y producing the same pair (x, f ( x , y)) is a signature. In fact, we will show that if for a 4-partition of [n] obtained as (x, f ( x , y)) for some x E and y ~ {0, 1, 2} no ternary sequence y is a signature, then there exist two sequences, z and w in such that x is different from both z and w and lies between them on a line which is a contradiction. This will imply that the number of signatures is at least one-fourth of that of all the pairs (x, y) with x E Ck, whence

Ck,

Ck

Ck

Ck,

[Ck[

m

~< 1{y; yis asignature[ ~< l

m

'

and thus

~<4(n + 1)2 max

min

where the last inequality follows by the usual entropy estimates for the binomial coefficients involved, cf. Section 1.2 in Csisz~tr and K6rner [5]. Choosing 2 = K/2,/~ = (1 - x)/2, this can be continued as

l l°g]Ckl<~21°g(n+l)+2

-

F/

H

[

~-max h(K/2)+ *¢~<1

(--2)(2~--~K)I 1 h

h(x/2)+(1-x/2)h((1-K)/(2-K)))

Noting that is nothing but the entropy of the probability distribution (K/2, 1 / 2 1/2) and thus equals 1 + i h(K), we conclude that it is not more than 3/2. But then lim sup -1 log

A(n) <~~.

x/2,

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121

It remains to prove that, in fact, if a 4-partition of [ n ] compatible with a sequence x ~ Ck has no signature, then to this sequence we can find two others in Ck so that the three lie on the same line. Let x be such a sequence and let us denote by A and B the two classes of the 4-partition which are contained in the set of the those coordinates where x is zero. Likewise, let C and D denote the remaining classes of the 4-partition, those whose elements are the coordinates where x is 1. Suppose that the ternary sequence y, defined by Yi = 0 if i s A, Yi = 1 if i 6 D, and Yi = 2 otherwise, is not a signature. This can only happen because there is a binary sequence z ~ Ck leading to the same ternary sequence y, implying that xi = zi whenever i ~ A w D. Further, there must be some coordinate in which x and z differ. Next we define a new ternary sequence y*, associated with the fixed 4-partition of x, as follows: Let y* equal 2 if l e a •D, and let y* equal xi in the other coordinates. By our hypothesis, y* cannot be a signature of x. This means that there exists a sequence w s C~ also leading to y*. Clearly, we have w~ = x~, whenever i e B u C. But x is equal to z in the coordinates belonging to A w D, proving that x lies between z and w which establishes the desired contradiction. Next we turn to the lower bound on ~. To obtain it, set n = 2k and consider the set Bn of all n-length binary sequences with Z~= ix, "= k. Let us choose M-element collections (i.e., M not necessarily distinct elements of Bn) randomly and equiprobably among all. Let us fix a sequence x e Bn and let E(n) denote the number of ordered pairs (y, z) ~ B ] such that x, y, z are on the same line, with y between x and z. This number is upper bounded as

E(n) <. 3 n. This means that the expected number of triples not in general position in our M-element collection will be more than

3

22n .

The right-hand side of this is not more than M/2, provided that 2n

M~<--

Thus, by choosing M as the largest integer satisfying the last inequality we obtain a collection of binary sequences of length n with the property that the number of undesirable triples of its elements is less than M/2. But

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JANOS KORNER

dropping all the sequences involved in any of these triples we obtain a set A of points in the Hamming space in general position with 2~ A ~>(3)n/2. This proves that lim n-. inf ~ 1n log A ( n ) >1 1 - ~1 log 3 = 0.21 ....

|

6. THE EVOLUTIONOF SETS Given three points x, y, and z in the n-dimensional Hamming space, we have been and will continue to be interested in the 3-length sequences appearing in their coordinates. To speed up the presentation, let us denote by W(x,y,z) the set of those elements aE {0, 1} 3, for which there is a coordinate i with (Xi, Yi, Zi) = a.

Our interest is in large subsets C of {0, 1} n with the property that for every three different elements x, y and z of C, the set W(x, y, z) contains an element of one or more specified subsets of {0, 1} 3. For brevity, we index the elements of {0, 1} 3 with the natural numbers of which they are the binary expansion, so that, e.g., (0, 0, 1) will be denoted by al, whereas (1,0, 1) will be written as as. A problem of our kind can now be formulated by specifying a family ~- of subsets of { 0, 1} 3, in the sense that given such a family we are interested, for every n and particularly, in the asymptotics of the maximum number N(~-, n) of elements in a subset C of {0, 1} n having the property that W(x, y, z) c~A # ~

for

x, y, z e C,

(7)

for every A e J . Let us denote 1

c~(~-) = lim sup -- log N(~-, n). n ----~ o o

n

Then we have the following known and new problems and results. PROBLEM ES [9].

Determine

a({ax, a2, a4}).

In particular, it is not known whether this number is less than 1. This problem has an equivalent formulation concerning delta systems

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123

Determine o~({al, a2}, {aa, a4}, {al, a4} ).

PROBLEMEK [10].

A set of sequences in {0, 1 } n having the property underlying the above definition is the set of characteristic vectors of a cancellative family of subsets of In] in the sense of Frankl and Fiiredi [ 10]. It is known that the in question is between log 3/3 and log 3 - 1, and Erd6s and Katona conjecture that it actually equals log 3/3. PROBLEME F F D R [7, 63.

Determine ~({al}, {a2}, {a4}).

Clearly, this 0~is just the quantitiy called y in the first part of this paper. PROBLEM F G U

[123.

Determine o¢({al, a6} , {82, a5} , {a4, a3} ).

This ~ is nothing but the ~ of the first part. It is needless to say that the quantity g in the asymptotics of qualitatively independent tripartitions also is an 0~, that of the family consisting of all 1-element subsets of {0, 1} 3. Yet we prefer to insert it into a different series of problems. A slightly different type of condition simply puts some restriction on the cardinality of the sets W(x, y, z). Let us denote

v(k) = ~ ( ~ ) = lim sup 1_log N(Yk, n), n~oo

n

where ~ is the family of all the subsets of {al, a2, a3, a4, a5, a6} having 7 - k elements. A more natural and in fact simpler way of putting this is to say that v(k) is the asymptotic exponent of the cardinality of the largest set C _ {0, 1 } " with the property that for every triple x, y, z of different elements of C the set W(x, y, z) c~ {al, a2, a3, a4, as, a6} has cardinality at least k. Obviously, v(1) = v(2)= 1. We have v(6)= g. The intermediate values of v(j), 3 ~l, while [W(x, y, z) c~ T[ >~m for every x s C, y~ C, z e C. Set

v(l, m) = lim sup _1log N(l, m, n). n~oo

n

While it is perfectly clear that ~({al}, {a2}, {a4})~
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JANOSKORNER

PROPOSITION 1. v(2, 2) = ½. Proof Consider the set C ' = {0, 1 } n/2, for n even, and for every x ~ C' define xn/2 + i = 1 - xi, 1 <<.i <~ n/2. The set containing all the n-length sequences so obtained, C___ { 0, 1 } ", has the desired property and thus establishes

v(2, 2) ~> ½. To prove the upper bound, consider a set C _ { 0, 1 } n achieving the maximum in the definition of N(2, 2, n) and fix an arbitrary sequence x ~ C. Without loss of generality, we can suppose that the number of l's in x is at most n/2. In fact, this can be guaranteed, since by complementing every sequence in C one obtains a new set that satisfies our condition, provided that it has been satisfied by C. Now, just note that by our condition any pair of the remaining sequences must differ in at least one of those coordinates in which x has a 1. Thus, N(2, 2, n) = IcI ~ 2 "/2. | In this paper we have singled out a class of problems involving excluded ternary configurations of binary sequences. In more complex problems the excluded configurations are not just ternary but k-ary, for k arbitrary, and the sequences need not be binary, either. Many of these more general problems are well-known and have an extended literature. The interested reader is referred to the recent survey article [ 31 ].

REFERENCES 1. N. ALON, Explicit construction of exponentially sized families of k-independent sets, Discrete Math. 58 (1986), 1991 193. 2. N. ALON, O. GOLOl~ICn, J. H.&STaD, AND R. PERa.LTA,Simple constructions of almost k-wise independent random variables, Proc. 1EEE-FOCS (1990), 544-553. 3. B. BBCI~R AND H.-U. SIMON, How robust is the n-cube, Inform. and Comput. 77 (1988), 162-178. 4. G. CottoN, J. K6RNER, AND G. SIMONY~,Zero-error capacities and very different sequences, in "Sequences, Combinatorics, Security and Transmission, Advanced International Workshop, Positano, Italy, June 1988" (R. M. Capocelli, Ed.), Springer, New York, 1990. 5. I. Csisz~.R AND J. KORNER, "Information Theory: Coding Theorems for Discrete Memoryless Systems," Akad6miai Kiad6, Budapest, 1981/Academic Press, New York, 1982. 6. A. G. DYACnKOVAND V. V. RYKOV, Bounds on the length of disjunctive codes, ProbL Per. Inform. 3 18 (1982), ~13. [in Russian]

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