C. E. Shannon, \The Zero Error Capacity of a Noisy Channel." IEEE Trans. on Information Theory, Vol 2, pp. S8-S19, September 1956.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for special classes of graphs. In this paper we provide the explicit formula for the Lov asz number of the union of two cycles, in two special cases, and a practically efficient algorithm, for the general case. 1 Introduction The notion of capacity of a graph was introduced by Shannon in [14], and after that labeled as Shannon capacity. This concept arises in connection with a graph representation for the problem of communicating messages in a zero error channel. One considers a graph G, whose vertices are letters from a given alphabet, and where adjacency indicates that two letters ....

C.E. Shannon. The zero-error capacity of a noisy channel, IRE Trans. Inform. Theory, IT-2 (1956), 8--19.

....and by forming arbitrary words composed of those two letter words we see that there are at least 5 #n 2# n symbol error free messages, an improvement over the earlier bound. An asymptotic study of the bit per symbol error free transmission rate leads to the notion of Shannon capacity of a graph [36], as follows. Let G be the graph with vertex set # and edge set al..l pairs of unconfoundable elements of #. Thus we are interested in maximum cliques of G , where G is the graph with vertex set al..l t tuples of vertices of G in which two such t tuples are adjacent if and only if for some ....

C. E. Shannon, The zero error capacity of a noisy channel, I.R.E. Trans. Inform. Theory IT-2 (1956), 8--19.

....8905 48164. A Sound Channel anagram of Claude Shannon 1 Introduction memory and feedback. The problem of optimal channel coding goes back to Shannon s original work. Sha1] The channel coding problem with feedback goes back to early work by Shannon, Dobrushin, Wolfowitz, and others. [Sha2], Dob1] Wol] Because of increased demand for wireless communication and networked systems there is a renewed interest in this problem. Feedback can increase the capacity of a noisy channel, decrease the complexity of the encoder and decoder, and reduce latency. There are ve main contributions ....

C. Shannon, \The Zero Error Capacity of a Noisy Channel," Institute of Radio Engineers, Transactions on Information Theory, Vol. IT-2 (September, 1956), pp. S8S19.

....noises and is an upper bound for all kinds of additive noises. However, in an environment with finite states and bounded noises, transmission error can actually be zero, instead of approaching zero as in Eq. 1) This motivated a research of zero error capacity initialed by Shannon in 1956 [11]. Quantization, if an upper bound on the quantization step exists, is an exampie of such a noise. We can find the zero error capacity of a digital image if quantization is the only source of distortion such as in JPEG. A braod study of theoretical watermarking capacity based on the above four ....

....on multivariant capacity analysis and four HVS models. In this paper, we focus on the zero error capacity of digital images. Shannon defined the zero error capacity of a noisy channel as the least upper bound of rates at which it is possible to transmit information with zero probability of error [11]. In contrast, here we will show that rather than a probability of error approaching zero with increasing code length, the probability of error can be actually zero under the conditions described above. This property is especially needed in applications that no errors can be tolerated. For ....

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C. E. Shannon, "The Zero-Error Capacity of a Noisy Chan- nel," IRE Trans. on Information Theory, IT-2: 8-19, 1956.

....2002. tATacT Bell Laboratories, 600 Mountain Ave. Murray Hill, NJ 07974, and Raymond and Beverly Sacklet Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. ATacT Bell Laboratories, 600 Mountain Ave. Murray Hill, NJ 07974. receives an arbitrary output in Sx. Folloxving Shannon [19], we study the amount of information a channel can communicate without error. Associated with a channel C is a characteristic graph . Its vertex set is X and two (distinct) vcr[iccs arc cmmcc[cd if [heir fan out sets interseel, namely, both can rcsuk in hc same ou[pu[ No[e ha[ every graph (V, E) ....

C. E. Shannon. The zero-error capacity of a noisy channel. IRE Transactions on Ivformation Theory, 2(3):8 19, 1956.

....[16] as well as of strong products of several cycles [17] In the second application we determine the independence numbers of the strong products C 7 2 Theta C 7 2 Theta C 2k 1 . Studies of the independence number of the strong product of odd cycles have been inspired by the Shannon s work [23] on the determination of the zerocapacity of a noisy channel. Shannon formulated the problem in the form of graph theory and supplied some partial results. It turns out that the solution of the problem requires the determination of the independence number of the strong product of graphs containing ....

C.E.Shannon, The zero-error capacity of a noisy channel, IRE Trans. Inform. Theory 2 (1956) 8--19.

....increase the capacity of the physically degraded broadcast channel, i.e. broadcast channels for which . It was later shown by Dueck [29] and Ozarow [80] 81] that feedback can in fact increase the capacity of general broadcast channels, in contrast to the single user channel, where Shannon [91] proved that feedback does not increase capacity. Ozarow and Leung [81] showed a new way to achieve the capacity region for the Gaussian broadcast channel with feedback using the Kailath Schalkwijk coding scheme, in which one uses feedback to attempt to correct the misperceptions of as seen by ....

C. E. Shannon, "The zero-error capacity of a noisy channel," IRE Trans. Inform. Theory, vol. IT-2, pp. 8--19, 1956.

....one can download paper [16] as well as other papers by Noam Nisan on communication complexity. From http: www.cs.yale.edu lovasz survey.html one can download paper [13] 1. 1 Shannon capacity and theta function The notion of capacity of a graph has been introduced by Shannon in [21], and after that was labeled as Shannon capacity. This concept arises in connection with a graph representation for the problem of communicating messages in a channel. One considers a graph G, whose vertices are letters from a given alphabet, and where adjacency indicates that two letters can be ....

....a review paper that should guide the study, and give an orientation for the analysis of the other papers. Concerning Shannon capacity and theta function, Donald Knuth has written a very nice and clear review [9] which is actually self contained and complete at the point that the other references ([21, 11]) have been included mostly for completeness and for providing the students with the original papers by Shannon and Lov asz. Matrix rigidity has been introduced in [22] A survey of both results and open questions is [7] References [5, 17, 2] contain results on properties of low rank matrices ....

C.E. Shannon. The zero-error capacity of a noisy channel, IRE Trans. Inform. Theory, vol. IT-2 (3), pp. 8--19 (1956).

.... As easily checked, the codebook C A n of an optimal code is precisely a maximal independent set of G(W n ) Consequently, the zero error capacity C 0 (W ) of channel W is equal to the capacity of the corresponding confoundability graph G(W ) C 0 (W ) C Gamma G(W ) Delta The paper [19] which Shannon published in 1956 and which contains these results inaugurated zero error information theory. Observe however that the last equality gives no real solution for the problem of assessing the zero error capacity of the channel, but simply re phrases it in a neat combinatorial language; ....

C. E. Shannon: "The Zero-Error Capacity of a Noisy Channel", IRE Trans. Inform. Theoryl, IT-2, 1956, 8-19

....be used to transmit data with an arbitrarily small probability of error. 1] C = supfRj8ffl 09N; E R N ; D R N P error (E R N ; D R N ) fflg Definition 2.3 The Shannon zero error capacity C of a channel is the maximal rate at which the channel can be used to transmit data without error. [8] C 0 = supfRj9N; E R N ; D R N P error (E R N ; D R N ) 0g In both definitions of capacity, the encoder decoder pair E R N ; D R N has rate R and end to end delay less than or equal to N . By inspection, we know C 0 C. Definition 2.4 The binary erasure channel has A = f0; 1g, B = ....

C. Shannon, "The Zero Error Capacity of a Noisy Channel." IEEE Trans. on Information Theory, Vol 2, pp. S8-S19, September 1956.

....and is an upper bound for all kinds of additive noises. However, in an environment with finite states and bounded magnitude noises, transmission error can actually be zero, instead of approaching zero as in Eq. 1. 1) This motivated a research of zero error 20 capacity initialed by Shannon in 1956 [117]. Quantization, if an upper bound on the quantization step exists, is an example of such a noise. We can find the zeroerror capacity of a digital image if quantization is the only source of distortion such as in JPEG. These difficulties motivated our work in Chapter 5. Avoiding to directly apply ....

....is the only source of distortion such as in JPEG and the largest applicable quantization steps are determined in advance. Shannon defined the zero error capacity of a noisy channel as the least upper bound of rates at which it is possible to transmit information with zero probability of error [117]. In contrast, here wewillshow that rather than a probabilityof error approaching zero with increasing code length, the probability of error can be actually zero under the conditions described above. This property is especially needed in applications that no errors can be tolerated. For instance, ....

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C. E. Shannon, "The Zero-Error Capacity of a Noisy Channel," IRE Trans. on Information Theory, IT-2: 8-19, 1956.

....receiver uses the return channel to inform the transmitter what letters were actually received; these letters are received at the transmitter before the next letter is transmitted and therefore can be used in choosing the next transmitted letter. In previous related work, Shannon rst proved in [16] that feedback does not increase the capacity of discrete memoryless channels (DMC s) He also conjectured that feedback can increase the capacity of channels with memory. 10 In a seemingly di erent work [11, 12] Jelinek investigated the capacity of nite state indecomposable channels with side ....

C. E. Shannon, \The Zero-Error Capacity of a Noisy Channel," IRE Transactions on Information Theory, Vol. 2, pp. 8-19, 1956.

....for Gg: In fact there are five equivalent definitions for Lov asz s theta function. The above definition is the definition # 2 (G) as described in [68, Equation (9.3.9) in page 287] and [100, Definition (6. 3) in page 9] Lov asz [106] introduced this function to approximate the Shannon capacity [140, 135, 115] of a graph. Clearly the theta function of G can be written as a semidefinite program as follows. min s.t. A ii = 1 for every i = 1; n A ij = 1 for every ij 62 G I Gamma A 0: Therefore #(G) is a polynomial time computable function. Let G be the complement graph of G. Clearly #( ....

C. E. Shannon. The zero-error capacity of a noisy channel. IRE Transaction on Information Theory, IT-2(3):8--19, 1956.

.... p 5) 2 and state the case k = 3 as an open problem. In this paper, we determine the limit (2) for every k, cf. Corollary 5) One feels that the above two problems have the same mathematical flavor. In a sense, they are close to the zero error capacity problems of Shannon s information theory [19] [13] In fact, in both cases the upper bounds can be derived using elementary information theoretic arguments. The corresponding lower bounds are not so easy to obtain. Our present results will be immediate consequences of a rather general theorem concerning Sperner capacities , a concept ....

C. E. Shannon, "The zero-error capacity of a noisy channel", IRE Trans. on Information Theory, 2, (1956), pp. 8--19.

....products are one of the most basic (but also least understood) operations in mathematics. Some well known problems dealing with the Cartesian product of graphs have been discussed in literature. First let us mention the famous Shannon problem concerning the zero error capacity of a noisy channel ([13]) The zero error capacity C 0 of a noisy channel is defined as the least upper bound of rates at which it is possible to transmit information with zero probability of error. Shannon also considered products of channels. The input alphabet of the product of two channels consists of all ordered ....

....Thus the matching number equals the packing number, in which the underlying hypergraph really is a graph and the disjoint sets of E are the independent edges. At the end of this historical sketch we return to the Shannon capacity. In [4] Alon disproved a conjecture of the work mentioned already, [13]. Shannon proved in [13] that for every two graphs G and H the inequality Theta(G [H) Theta(G) Theta(H) holds and that equality holds for graphs with special properties. Shannon remarked in [13] We conjecture but have not been able to prove that the inequalities in Theorem 4 hold in ....

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C.E.Shannon, The zero-error capacity of a noisy channel, IRE Trans. Inform. Theory 2, 8-19, 1956.

....V ae X is said to be an internally stable set of G if V GammaV = This implies that no element of V has a loop. We would like to call a set J ae X with no 2 vertices joined by an edge a Shannon stable or briefly S stable set of a graph, because this concept has been used by Shannon in [1] and because the difference between the two notions of stability seems not to have been emphasized enough in the literature even though it is significant for product graphs. In an S stable set elements with loops are permitted. An internally stable set is S stable. The converse is not ....

....Choose E as set of vertices and join E; E 0 2 E by an edge if and only if E E 0 6= G(H) is a graph with all loops in the edge set and the packings of H n are in one to one correspondence to the S stable sets of G(H) n . C. Shannon s zero error capacity. Problem 2. is due to Shannon [1]. It is a graph theoretic formulation of the information theoretic problem of determining the maximal number of messages which can be transmitted over a memoryless noisy channel with error probability zero. lim n 1 1 n log j(G n ) was called in [1] the zero error capacity C o , say. Using our ....

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Shannon, C.E., The zero--error capacity of a noisy channel. IRE Trans. Inform. Theory, IT--2, 8--19, 1956.

.... limit exists, by super multiplicativity, and it is always at least ff(G) It is worth noting that it is sometimes customary to call log c(G) the Shannon capacity of G, but we prefer to use here the above definition, following Lov asz [12] The study of this parameter was introduced by Shannon in [14], motivated by a question in Information Theory. Indeed, if V is the set of all possible letters a channel can transmit in one use, and two letters are adjacent if they may be confused, then ff(G n ) is the maximum number of messages that can be transmitted in n uses of the channel with no ....

....Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Part of this work was done at DIMACS and at the Institute for Advanced Study, Princeton, NJ 08540, USA. Research supported in part by a grant from the the Israel Science Foundation. Email: noga math.tau.ac. il Shannon [14] proved that for every G and H , c(G H) c(G) c(H) and that equality holds if the vertex set of one of the graphs, say G, can be covered by ff(G) cliques. He conjectured that in fact equality always holds. In this paper we prove that this is false in the following strong sense. Theorem 1.1 For ....

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C. E. Shannon, The zero--error capacity of a noisy channel, IRE Trans. Inform. Theory 2 (1956), 8--19.

....is never connected to itself. 1.1 Channel coding A channel consists of a finite input set X , an output set Y , and a nonempty fan out set S x Y for every x 2 X . In each channel use, a sender transmits an input x 2 X and a receiver receives an arbitrary output in S x . Following Shannon [15], we study the amount of information a channel can communicate without error. Associated with a channel C is a characteristic graph G. Its vertex set is X and two (distinct) vertices are connected if their fan out sets intersect, namely, both can result in the same output. Note that every graph ....

C. E. Shannon. The zero-error capacity of a noisy channel. IRE Transactions on Information Theory, 2(3):8--19, 1956.

.... cover of a graph are equivalent and are known to be NP complete [7] Practical applications of these models are abundant, e.g. information retrieval, signal transmission analysis, classification theory, economics, scheduling, experimental design and computer vision (See [1] 2] 3] 5] 6] [15], 9] and [16] for details) A simple integer programming formulation for vertex packing is given next. Let S ae V be an independent k set of the graph G = V; E) and define w j = 8 : 1 if vertex j 2 S Gamma1 otherwise A ( 1,1) integer programming formulation is: Find w 2 jV j such ....

C.E. Shannon. The zero-error capacity of a noisy channel. I.R.E. Transactions, 3, 1956.

....which is proved in Section 12. Theorem 3. a) Cm de f (W 0 ) 8 : max PXS I(XS Y ) or 0 (b) Cm de f (W 0 ) 0 exactly if all columns have positive or zero entries only. The astute reader may notice that Shannon s formula or the alternate formula of [15] asked for by Shannon in [8]) has also a dichotomy relative to positivity. The formula in [15] describes the capacity of a jamming problem, namely that of an arbitrarily varying channel with feedback. Our formula for Cm de f (W 0 ) also settles a feedback problem involving jamming. Problem 1. Is there a common ....

C.E. Shannon, "The zero--error capacity of a noisy channel", IRE--IT 2, 8--19, 1956.

....vertices in the pentagon graph C 5 . More generally, the nth power of P is equivalent to finding the largest independent set of vertices in the nth strong power of C 5 (see [4] for graph product and graph power terminology) The limit n q v(P ) is known as the Shannon capacity of the graph C 5 [9]. It has been shown [5] that the Shannon capacity of the graph C 5 is p 5; on the other hand, v 3 (P ) is 5=2, since (1=2; 1=2; 1=2; 1=2; 1=2) is a solution to both P and P . This example shows that Theorem 1 does not dualize to a theorem about maximization programs; that is, n q v( P ....

C. E. Shannon, The zero-error capacity of a noisy channel, IRE Trans. Info. Theory IT-2 (1956), 8-19.

....; y n 2 f0; 1; 2g n for some t 2 [n] fx t ; y t g = f0; 1g: 9.2) One readily verifies that jA j 2 n and equality occurs for A = f0; 1g n . In fact the problem is equivalent to Shannon s zero error capacity problem for the matrix 0 1 0 0 1 1 2 1 2 1 A . As Shannon noticed in [11], it equals log 2 2 = 1 . ....

C.E. Shannon, "The zero--error capacity of a noisy channel", IRE Trans. Inform. Theory, vol. IT--2, no. 3, 8--19, 1956.

....Y , and transmission matrix W . By adding letters, if necessary, we can always assume that X ae Y . Recall that for two words x n 2 X n and y n 2 Y n W n (y n jx n ) n Y t=1 W (y t jx t ) 1. 1) Our studies are devoted to cases with zero error probabilities for decisions (see [1]) They concern the performance of this channel for transmission codes under two criteria, namely, the erasure probability and the average list size. We also introduce identification codes with zero error probability for misrejection. Let us fix any blocklength n . A code C for the channel is ....

C.E. Shannon, "The zero error capacity of a noisy channel", IRE--IT2, 8--19, 1956.

....subgraph is often called a clique and correspondingly, the cardinality of the largest clique of G is called its clique number. The analysis of its asymptotic growth in large product graphs leads to one of the most formidable problems in modern combinatorics. This problem was posed by Shannon [14] in 1956 in connection with his analysis of the capability of certain noisy communication channels to transmit information in an error free manner. Shannon associated a graph with every channel. In our notation (which is different from his) the vertex set of the graph represents the symbols that ....

....the receiver. Formally, x; y) 2 E(G n ) if and only if 9i (x i ; y i ) 2 E(G) where we suppose that x i and y i are the i th coordinate of x and y, respectively. As usual, E(G) denotes the edge set of the graph G. Following Berge [1] G n is called the n th co normal power of G. Shannon [14] observed that if K is a clique in G then K n is a clique in G n , whence the clique number of G n is at least as big as the n th power of the clique number of G. In fact, these two quantities coincide whenever the clique number of G equals its chromatic number. This observation led Berge ....

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C. E. Shannon, The zero--error capacity of a noisy channel, IRE Trans. Inform. Theory, 2(1956), 8--19

....y AT T Bell Laboratories, 600 Mountain Ave. Murray Hill, NJ 07974, and Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. z AT T Bell Laboratories, 600 Mountain Ave. Murray Hill, NJ 07974. receives an arbitrary output in S x . Following Shannon [19], we study the amount of information a channel can communicate without error. Associated with a channel C is a characteristic graph G. Its vertex set is X and two (distinct) vertices are connected if their fan out sets intersect, namely, both can result in the same output. Note that every graph ....

C. E. Shannon. The zero-error capacity of a noisy channel. IRE Transactions on Information Theory, 2(3):8--19, 1956.

....tell the one from the other and this may depend on the way we are looking at them. Shannon proposed to look at them through a noisy channel and thus has created two tremendously difficult problems for the combinatorialist: the code distance problem [14] 20] and the zero error capacity problem [26]. With both of these problems one is interested in how many pairwise really different sequences we can construct for given V and n. In the code distance problem we fix some ff 0 and say that v and v 0 2 V n are r. d. really different) if they differ in at least ffn positions, i.e. if ....

....to the communication engineer and has a vast literature. The problem of zero error capacity of the discrete memoryless channel is familiar to the graph theorist in a channel free formulation. The stochastic description of the channel is translated into purely graph theoretic terms. Shannon [26] defines an arbitrary graph G on the set V and makes us call two elements of V n really different if among the coordinate pairs (v 1 ; v 0 1 ) v n ; v 0 n ) of v = v 1 : v n and v 0 = v 0 1 : v 0 n there is some edge (a; b) 2 E(G) i.e. if we have (v i ; v 0 i ) 2 ....

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C. E. Shannon, "The zero-error capacity of a noisy channel", IRE Trans. Inform. Theory, 2, (1956), pp. 8--19.

.... n p (G n ) where the power of G is relative to either the disjunctive or lexicographic product of graphs (see [5] 10] 13] on the other hand, it is not true that F (G) always equals lim n 1 n p (G n ) In fact, this limit gives the Shannon capacity of the complement of G [14], which is known to be p 5 when G is the pentagon C 5 [9] while F (C 5 ) 5 2 ) THE GRAPH TRANSFORMATION OF MYCIELSKI Motivated by [11] given a graph G we define a graph (G) as follows. If G has vertex set fv 1 ; v 2 ; v m g, let V ( G) fx 1 ; x 2 ; xm ; y 1 ; y 2 ....

C. E. Shannon, The zero-error capacity of a noisy channel, IRE Trans. Info. Theory IT-2 (1956), 8-19.

....that is the minimal number of sets from E n to cover V n , have been studied in the literature, this seems to be not the case for the partition number (n) Obviously, c(n) n) p(n) if c(n) and (n) are well defined. The quantity lim n 1 1 n log p(n) is Shannon s zero error capacity ([4]) Whereas it is known only in very few cases (see [5] for lim n 1 1 n log c(n) a nice formula exists (see [6] 7] The difficulties in analyzing (n) are similar to those for p(n) For the case of graphs with edge set E including all loops we prove that (n) 1) n (Theorem 3) This ....

....set of all probability distributions on E , q E is the probability of E under q and 1 E is the indicator function of the set E . Example 4. V 4 = f0; 1; 2; 3; 4g , E 4 = Phi fx; x 1 mod 5g : x 2 V 4 Psi . Here we have 5 = p(H 4 Theta H 4 ) 6= p(H 4 )p(H 4 ) 4: 6. 3) It was shown in [4] that this is the smallest example in the previous sense. Notice that it is bigger than the previous one. Example 5. In order to avoid heavy notation we write H 5 = V 5 ; E 5 ) simply without an index as H = V; E) It is constituted by the 5 vertex sets W i = fx ij : j = 1; 2; mg; 3 ....

C.E. Shannon, The zero--error capacity of a noisy channel, IRE Trans. Inform. Theory IT-2 (1956), 8--19.

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C. E. Shannon, The zero-error capacity of a noisy channel, IRE Transactions on Information Theory, 2(3) (1956), 8-19. 11 Department of Mathematics Massachusetts Institute of Technology, Cambridge, MA 02139 E-mail address: tbohman@moser.math.cmu.edu Current address: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213

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C. E. Shannon, "The zero-error capacity of a noisy channel," IRE Transactions on Information Theory, 2(3) (1956), 8--19.

....of the Lov asz theta function. 1 Introduction The purpose of this note is to study the Shannon capacity of odd cycles, and give some insights into this strickingly difficult problem, which remains open even for the 7 cycle. The notion of capacity of a graph has been introduced by Shannon in [12], and after that was labeled as Shannon capacity. This concept arises in connection with a graph representation of the problem of communicating messages in a zero error channel. One considers a graph G, whose vertices are letters from a given alphabet, and where adjacency indicates that two ....

C.E. Shannon. The zero-error capacity of a noisy channel, IRE Trans. Inform. Theory, vol. IT-2 (3), pp. 8--19 (1956).

....The search was done by a computer program using the simulated annealing algorithm with a constant time temperature schedule. Keywords: Shannon capacity; Combinatorial problems; Design of algorithms. 1 Introduction and definitions The study of Shannon capacity was introduced by Shannon in [13], and motivated by the determination of the zero error capacity of a noisy channel. Shannon formulated the problem in the form of graph theory and supplied some partial results. It turns out that the solution of the problem requires the determination of the independence number of product of graphs ....

....the solution of the problem requires the determination of the independence number of product of graphs which contain odd cycles. The Shannon capacity of the 7 cycle is one of the unsolved problems given in [6] It may be interesting to note that the capacity of C 5 was studied already by Shannon [13] in 1956, and was determined only in 1979 by Lov asz [10] All graphs considered in this paper are connected finite undirected graphs without loops or multiple edges. If G is a graph, we shall write V (G) or V for its vertex set and E(G) or E for its edge set. E(G) is a set of unordered pairs xy ....

C.E.Shannon, The zero-error capacity of a noisy channel, IRE Trans. Inform. Theory 2 (1956) 8-19.

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C. E. Shannon, \The Zero Error Capacity of a Noisy Channel." IEEE Trans. on Information Theory, Vol 2, pp. S8-S19, September 1956.

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C. E. Shannon, "The Zero Error Capacity of a Noisy Channel." IEEE Transactions on Information Theory, Vol 2, pp. S8-S19, September 1956. 33

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C. E. Shannon, "The zero error capacity of a noisy channel," IEEE Trans. Inform. Theory, vol. 2, no. 3, pp. 8--19, Sept. 1956.

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C. E. Shannon, \The Zero Error Capacity of a Noisy Channel." IEEE Trans. on Information Theory, Vol 2, pp. S8-S19, September 1956.

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C. E. Shannon, "The zero error capacity of a noisy channel," IRE Trans. Inform. Theory, vol. IT-2, pp. 8--19, Sep. 1956.

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C. Shannon, "The Zero Error Capacity of a Noisy Channel." IEEE Trans. on Information Theory, Vol 2, pp. S8-S19, September 1956.

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C. Shannon, "The Zero Error Capacity of a Noisy Channel." IEEE Transactions on Information Theory, Vol 2, pp. S8-S19, September 1956. 57

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Shannon, C.E.: The Zero-Error Capacity of a Noisy Channel. IRE Trans. Inform. Theory, IT-2 (1956) 8-19

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Shannon, C.E., The zero-error capacity of a noisy channel, IRE Trans. Inform. Theory, IT-2 (1956) 8-19 20

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C. E. Shannon, The zero error capacity of a noisy channel, I.R.E. Trans. Inform. Theory IT-2 (1956), 8--19.

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C. E. Shannon, \The zero-error capacity of a noisy channel," IRE Trans. on Information Theory, vol. IT-2, pp. 8-19, September 1956.

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Shannon, C.E., The zero-error capacity of a noisy channel, IRE Trans. Inform. Theory, IT-2 (1956) 8-19 19

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Claude E. Shannon. The zero error capacity of a noisy channel. IRE Trans. on Information Theory, Vol. IT-2:S8--S19, September 1956.

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Shannon, C.E., The zero-error capacity of a noisy channel, IRE Trans. Inform. Theory, IT-2 (1956) 8-19 19

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C. E. Shannon, "The Zero-Error Capacity of a Noisy Channel, " IRE Trans. on Information Theory, IT-2: 8-19, 1956.

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C. E. Shannon, "The zero-error capacity of a noisy channel", IRE Trans. on Information Theory, 2, (1956), pp. 8--19.

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