Abstract: It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} . The method is then generalized to obtain upper bounds on the capacity of an arbitrary graph. A well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases. Several results are obtained on the capacity of special graphs; for example, the Petersen graph has capacity four and a self-complementary graph with n points and with a vertex-transitive automorphism group has capacity \sqrt{5} .

Abstract: The multiaccessing of a broadcast communication channel by independent sources is considered. Previous accessing techniques suffer from long message delays, low throughput, and/or congestion instabilities. A new class of high-speed, high-throughput, stable, multiaccessing algorithms is presented. Contentions resolving tree algorithms are introduced, and they are analyzed for specific probabilistic source models. It is shown that these algorithms are stable (in that all moments of delay exist) and are optimal in a certain sense. Furthermore, they have a maximum throughput of 0.430 packets/slut and have good delay properties. It is also shown that, under heavy traffic, the optimally controlled tree algorithm adaptively changes to the conventional time-division multiple access protocol.

Abstract: In 1948 W. R. Bennett used a companding model for nonuniform quantization and proposed the formula D \: = \: \frac{1}{12N^{2}} \: \int \: p(x) [ E(x) ]^{-2} \dx for the mean-square quantizing error where N is the number of levels, p (x) is the probability density of the input, and E \prime (x) is the slope of the compressor curve. The formula, an approximation based on the assumption that the number of levels is large and overload distortion is negligible, is a useful tool for analytical studies of quantization. This paper gives a heuristic argument generalizing Bennett's formula to block quantization where a vector of random variables is quantized. The approach is again based on the asymptotic situation where N , the number of quantized output vectors, is very large. Using the resulting heuristic formula, an optimization is performed leading to an expression for the minimum quantizing noise attainable for any block quantizer of a given block size k . The results are consistent with Zador's results and specialize to known results for the one- and two-dimensional cases and for the case of infinite block length (k \rightarrow \infty) . The same heuristic approach also gives an alternate derivation of a bound of Elias for multidimensional quantization. Our approach leads to a rigorous method for obtaining upper bounds on the minimum distortion for block quantizers. In particular, for k = 3 we give a tight upper bound that may in fact be exact. The idea of representing a block quantizer by a block "compressor" mapping followed with an optimal quantizer for uniformly distributed random vectors is also explored. It is not always possible to represent an optimal quantizer with this block companding model.

Abstract: A coding theorem for the discrete memoryless broadcast channel is proved for the case where no common message is to he transmitted. The theorem is a generalization of the results of Cover and van der Meulen on this problem. The result is tight for broadcast channels having one deterministic component

Abstract: How much separate information about two random binary sequences is needed in order to tell with small probability of error in which positions the two sequences differ? If the sequences are the outputs of two correlated memoryless binary sources, then in some cases the rate of this information may be substantially less than the joint entropy of the two sources. This result is implied by the solution of the source coding problem with two separately encoded side information sources for a special class of source distributions.

Abstract: The structure of punctured convolutional cedes is described, and it is indicated how their use simplifies the design of maximum likelihood decoders. The best codes of this class for rates 2/3 and 3/4 are tabulated and performance curves are given for these codes.

Abstract: In Part I of this three-part study, it was shown that novel quantum states, called two-photon coherent states (TCS), have significant potential for improving free-space optical communications. Because TCS radiation does not possess a classical analog, i.e., its diagonal P -representation is highly singular, the semiclassical conditional Poisson process model for direct detection is not applicable to TCS reception. In this paper, photoemissive detection of arbitrary quantized radiation fields is studied with incorporation of the nontrivial effects of detector quantum efficiency. General theorems are derived permitting the application of classical point process results to the detection and estimation of signals in arbitrary quantum states. These general theorems are applied to determining the performance of TCS optical communication systems that employ direct, heterodyne, or homodyne detection in binary decision as well as in linear modulation problems. It is shown that the use of TCS radiation with direct detection or heterodyne detection results in minimal performance increments over comparable coherent-state systems. Homodyne detection, however, can achieve the full TCS signal-to-noise ratio improvement predicted in Part I of this study. The increase in homodyne signal-to-noise ratio obtained by use of TCS radiation yields significant performance gains in both linear modulation and antipodal signal detection.

Abstract: Delsarte's linear programming bound (an upper bound on the cardinality of cliques in association schemes) is compared with Lov\acute{a}sz's \theta -function bound (an upper bound on the Shannon capacity of a graph). The two bounds can be treated in a uniform fashion. Delsarte's linear programming bound can be generalized to a bound \theta \prime(G) on the independence number \propto(G) of an arbitrary graph G , such that \theta \prime(G) \leq \theta(G) . On the other hand, if the edge set of G is a union of classes of a symmetric association scheme, \theta(G) may be calculated by linear programming, For such graphs the product \theta(G) . \theta(G) is equal to the number of vertices of G .

Abstract: In the discrimination problem the random variable \theta , known to take values in {1, \cdots ,M} , is estimated from the random vector X . All that is known about the joint distribution of (X, \theta) is that which can be inferred from a sample (X_{1}, \theta_{1}), \cdots ,(X_{n}, \theta_{n}) of size n drawn from that distribution. A discrimination nde is any procedure which determines a decision \hat{ \theta} for \theta from X and (X_{1}, \theta_{1}) , \cdots , (X_{n}, \theta_{n}) . For rules which are determined by potential functions it is shown that the mean-square difference between the probability of error for the nde and its deleted estimate is bounded by A/ \sqrt{n} where A is an explicitly given constant depending only on M and the potential function. The O(n ^{-1/2}) behavior is shown to be the best possible for one of the most commonly encountered rules of this type.

Abstract: The answers to several problems of Lov\hat{a}sz concerning the Shannon capacity of a graph are shown to be negative.

Abstract: A new series for the interpolation of band-limited functions is found by using an approximation to tho prolate spheroidal wave function as a convergence factor. A truncation error bound is found for the central interpolation of band-limited functions using the new series. It is shown that this bound is lower than other known bounds in many cases of interest.

Abstract: A two-stage iterative algorithm for selecting a subset of a training set of samples for use in a condensed nearest neighbor (CNN) decision rule is introduced. The proposed method uses the concept of mutual nearest neighborhood for selecting samples close to the decision line. The efficacy of the algorithm is brought out by means of an example.

Abstract: Algorithms are presented for encoding and decoding strings of characters as real binary fractions, using registers of fixed precision. The encoding is left to right and does not require blocking. The algorithms have storage requirements O(N) and computation time O(n \log_{2}N) for string length n and alphabet size N .

Abstract: Reservation and TDMA schemes are studied for governing the access-control discipline for a network of terminals communicating through a multi-access broadcast channel. A single repeater is employed to allow a fully connected network structure. A channel can be characterized as inducing a Iow propagation-delay value, as for terrestrial radio or fine networks, or as being associated with a higher propagation-delay value, as for a satellite communication channel. A synchronized (slotted) communication medium is considered. Messages are composed of a random number of packets, governed by an arbitrary message-length distribution. The process describing the number of reserved message arrivals within each time frame is assumed to be a sequence of i.i.d, random variables, governed by an arbitrary distribution. (A Poisson arrival stream thus becomes a special case.) The reservation access-control disciplines studied in this paper employ message-switching distributed-control procedures. The performance of each access-control scheme is evaluated according to its delay-throughput function. In particular, schemes are developed to adapt their structure, or protocol, dynamically to the underlying fluctuating network traffic-flow values. A fixed-reservation access-control (FRAC) discipline is studied, employing a fixed periodic pattern of reservation and service periods. The reservation periods are used for the transmission of reservation packets as well as for the integrated service of other groups of network stations. The latter stations can access the channel during these periods, using any proper access control procedure. As a special case, message-delay distributions and moments under a TDMA scheme are obtained. Using dynamic estimates of the underlying message traffic parameters, a dynamic fixed-reservation access-control (DFRAC) scheme is obtained. An analytical technique, which employs a Markov ratio limit theorem, is presented for the derivation of the delay-throughput performance curves of dynamic demand-assignment reservation schemes. To illustrate its application, asynchronous reservation demand-assignment (ARDA) schemes are developed to adapt automatically to the underlying network traffic characteristics. Such schemes establish reservation slots dynamically according to observed network service demands and queue sizes.

Abstract: A variety of theoretical results are derived for a well-known class of discrete-time adaptive filters. First the following idealized identification problem is considered: a discrete-time system has vector input x(t) and scalar output z(t)= h \' x(t) where h is an unknown time-invariant coefficient vector. The filter considered adjusts an estimate vector \hat{h}(t) in a control loop according to \hat{h}(t + \Delta t) = \hat{h}(t) + K[z(t) - \hat{z} (t)]x(t) , where \hat{z}( t)= \hat{h}( t) \' x( t) and K is the control loop gain. The effectiveness of the filter is determined by the convergence properties of the misalignment vector r(t) = h - \hat{h}(t) . It is shown that a certain nondegeneracy "mixing" condition on the Input { x(t)} is necessary and sufficient for the exponential convergence of the misalignment. Qualitatively identical upper and lower bounds are derived for the rate of convergence. Situations where noise is present in z(t) and x(t) and the coefficient vector h is time-varying are analyzed. Nonmixing inputs are also considered, and it is shown that in the idealized model the above stability results apply with only minor modifications. However, nonmixing input in conjunction with certain types of noise lead to bounded input - unbounded output, i.e., instability.

Abstract: Evidence is given for the difficulty of an eventual proof of computational security for cryptosystems based on one-way functions, such as the one proposed by Diffie and Hellman. A proof of NP-completeness for the cryptanalytic effort would imply NP=CoNP.

Abstract: The directed divergences of two probability densities p and q are given by \int p(x) \log (p(x)/q(x))dx and by the same expression with p and q interchanged; the divergence is the sum of the directed divergences. These quantities have applications in information theory and to the problem of assigning prior probabilities subject to constraints. It is shown that the directed divergences and their positive linear combinations, including the divegeoce, are characterized by axioms of \it{positivity, additivity}, and \it{finiteness}, which are fundamental in work on prior probabilities. In the course of the proof, the latter two are shown to imply yet another axiom: \it{linear invariance}.

Abstract: Tbe problem of transmission of separate messages to each of two receivers over a general binary-input broadcast channel is investigated. A new approach to a class of information-theoretic problems is developed and applied to obtain bounds on the cardinalities of auxiliary random variables. These bounds permit the calculation of two different regions of achievable rate pairs which are derived from the Cover-van der Meulen region {\cal R} of achievable rate triples. Numerical evaluation of these regions of rate pairs for two examples demonstrates that the region {\cal R} can be enlarged. This enlargement is accomplished by making {\cal R} internally consistent, as the true capacity region must be. The results display complex interactions between common and separate information in broadcast problems.

Abstract: Given a wide-sense forwards Markovian model, namely a state-space system driven by a white-noise process that is uncorrelated with the random initial state of the system, several papers have shown how to obtain a corresponding {\sl backwards} Markovian model: a system driven backwards in time from a 'terminal' state by a white-noise process that is uncorrelated with this state, and such that the joint second-order statistics of the backwards-model state and output processes equal those of the corresponding forwards-model processes. A very direct derivation is presented that shows in effect how a simple rearrangement of the forwards

Abstract: A class of time-invariant binary convolutional codes is defined, called partial-unit-memory codes. These codes are optimal in the sense of having maximum free distance for given values of R, k (the number of encoder inputs), and mu (the number of encoder memory cells). Optimal codes are given for rates R = 1/4, 1/3, 1/2, and 2/3, with mu not greater than 4 and k not greater than mu + 3, whenever such a code is better than previously known codes. An infinite class of optimal partial-unit-memory codes is also constructed based on equidistant block codes.

Abstract: An upper bound on the rate distortion function is obtained for source coding with partial side information at the decoder. Previous results were for complete side information, i.e. full knowledge of Y_{n} . below. A diagram given in the paper helps to describe the problem. The bound is given in

Abstract: The problem of designing memoryless detectors for known signals in stationary m-dependent noise processes is considered. Applying the criterion of asymptotic relative efficiency, the optimal such detector is shown to be characterized by the solution to a Fredholm integral equation whose kernel depends only on the second-order probability distributions of the noise. General expressions are derived for this solution and for the asymptotic efficiency of the optimal detector relative to other memoryless detectors. To illustrate the analysis, specific results are given for the particular case where the noise process is derived by memoryless nonlinear transformation of a Gaussian process. In addition, an extension of the analytical results to the more general case of \phi -mixing noise processes is discussed.

Abstract: A recent paper [1] discussed the 2^{-p} bound (where p = n- k ) for the probability of undetected error P(\epsilon) for an (n,k) block code used for error detection on a binary symmetric channel. This investigation is continued and extended and dual codes are studied. The dual and extension of a perfect code obey the 2^{-p} bound, but this is not necessarily true for arbitrary codes that obey the bound. Double-error-correcting BCH codes are shown to obey the bound. Finally the problem of constructing uniformly good codes is examined.

Abstract: A natural and easily verifiable sufficient condition is given for the nonergodicity of a Markov chain. The result provides a converse to a theorem of Pakes and implies as special cases several recently published sufficient conditions for the instability of various forms of slotted ALOHA systems with Markovian retransmissions.

Abstract: A new class of time-invariant binary convolutional codes is defined, called partial-unit-memory codes. These codes are optimal in the sense of having maximum free distance for given values of R, k (the number of encoder inputs), and \mu (the number of encoder memory cells). New optimal codes are given for rates R=l/4, 1/3, 1/2, \and 2/3 with \mu \leq 4 \and k \leq \mu + \3 , whenever such a code is better than previously known An infinite class of optimal partial-unit-memory codes is also constructed based on equidistant block codes.

Abstract: A method for evaluating the average error probability in digital fiber optic transmission systems is described. It takes into account the statistical properties of the main impairments that affect the information signal, i.e., shot noise (dark current included), thermal noise, and intersymbol interference. Unlike other methods, the analysis of shot-noise statistics does not entail the second-moment characterization alone, but uses a number of moments sufficient to evaluate the error probability with a high degree of accuracy. For this purpose it is shown how the cumulants of shot noise and intersymbol interference of any order can be computed. The baseband pulse response and the statistical distribution of the avalanche-photodetector gain are assumed to be available. Moreover, techniques for exploiting the cumulants are given; knowledge of the cumulants allows one to compute the probability of error by means of Gaussian numerical integration rules. Finally, some results and comparisons are reported.

Abstract: Quantizers for digital coding systems are usually optimized with respect to a model of the probability density function of the random variable to be quantized. Thus a mismatch of the quantizer relative to the actual statistics of the random variable may be unavoidable. This paper presents the results of an experimental investigation of three types of mismatch. For the modeling of the source statistics, the gamma-, Laplacian-, and Gaussian-distribution are used. The optimization of the quarttizers is carried out with respect to the minimum mean-square error criterion.

Abstract: A class of nonlinear receiver structures is described for the detection of weak signals in non-Gaussian narrowband noise. In particular, the concept of a locally optimum receiver structure is extended to the ease of narrowband signal and noise models. A useful class of non-Gaussian narrowband noise models is developed for which the locally optimum receiver implementation is explicitly determined. These structures are shown to provide considerable improvement over conventional linear receiver structures. The basis of comparison is taken as the asymptotic relative efficiency (ARE). Unfortunately, the locally optimum receiver requires explicit {sl a priori} knowledge of the underlying noise distribution. To circumvent this difficulty a rather simple adaptive nonlinear receiver structure is described which attempts to adapt to the unknown prevailing noise environment. This adaptive receiver is shown to provide fairly efficient and robust performance in a wide variety of non-Gaussian narrowband noise environments.

Abstract: A method for constructing Lee metric codes over arbitrary alphabet sizes using the elementary concepts of module theory is presented. The codes possess a high degree of symmetry. Codes with two information symbols over arbitrary alphabet sizes are cyclic reversible. For alphabet sizes which are a power of two or an odd prime number, codes with one information symbol are reversible and equidistant, and codes having more than two information symbols are quasi-cyclic reversible. Binary Reed-Muller codes arise as subcodes of the codes presented. A method of constructing equidistant Lee metric codes analogous to maximum length shift register codes is presented.