The number of monomials required to interpolate a received word in an algebraic list decoder for Reed–Solomon codes depends on the instantaneous channel error, and not only on the decoder design parameters. The implications of this fact are that the decoder should be able to exhibit lower decoding complexity for low-weight errors and, consequently, enjoy a better average-case decoding complexity and a higher decoding throughput. On the analytical side, this paper studies the dependence of interpolation costs on instantaneous errors, in both hard- and soft-decision decoders. On the algorithmic side, it provides an efficient interpolation algorithm, based on the state-of-the-art interpolation algorithm, that enjoys reduced running times for reduced interpolation costs.

A sphere decoder searches for the closest lattice point within a certain search radius. The search radius provides a tradeoff between performance and complexity. We focus on analyzing the performance of sphere decoding of linear block codes. We analyze the performance of soft-decision sphere decoding on AWGN channels and a variety of modulation schemes. A hard-decision sphere decoder is a bounded distance decoder with the corresponding decoding radius. We analyze the performance of hard-decision sphere decoding on binary andq-ary symmetric channels. An upper bound on the performance of maximum-likelihood decoding of linear codes defined overFq(e.g. Reed- Solomon codes) and transmitted overq-ary symmetric channels is derived and used in the analysis. We then discuss sphere decoding of general block codes or lattices with arbitrary modulation schemes. The tradeoff between the performance and complexity of a sphere decoder is then discussed.

A fundamental function performed by a sensory network is the retrieval of data gathered collectively by sensor nodes. The metrics that measure the efficiency of this data collection process are time and energy. In this paper, we study via simple discrete mathematical models, the statistics of the data collection time in sensory networks. Specifically, we analyze the average minimum delay in collecting randomly located/distributed sensors data for networks of various topologies when the number of nodes becomes large. Furthermore, we analyze the impact of various parameters such as size of packet, transmission range, and channel erasure probability on the optimal time performance. Our analysis applies to directional antenna systems as well as omnidirectional ones. This paper focuses on directional antenna systems and briefly presents results on omnidirectional antenna systems. Finally, a simple comparative analysis shows the respective advantages of the two systems.

A binary block interference channel (BIC) is model of binary channels with memory that allows for a mathematically tractable computation of channel capacity. One can easily imagine interconnecting such channels into a network that allows point-to-point communication between any two nodes in the network. Given a pair of network nodes, finding the path with the highest capacity is quite trivial if we can assume that all participating nodes in any path connecting the two nodes can perform coding at arbitrary complexity such that at each link capacity is achieved. However, even if the complexity assumption is not taken into account, in most real-life networks (such as the current Internet), only a minimum amount of coding is performed at the link layer. In most networks, coding is performed five or six layers up in the OSI network model, i.e., on either the presentation or the application layer. Under such realistic circumstances, finding the path with the highest capacity is no longer trivial. In this paper, we propose a solution based on a modified version of the Dijkstrapsilas Algorithm.

In this paper, we extend the optimal worst-case QoS routing algorithm and metric definition given in [1]. We prove that in addition to the q-ary symmetric and q-ary erasure channel model, the necessary and sufficient conditions defined in [2] for the Generalized Dijkstra's Algorithm (GDA) can be used with a constrained non-negative-mean AWGN channel. The generalization allowed the computation of the worst-case QoS metric value for a given edge weight density. The worst-case value can then be used as the routing metric in networks where some nodes have error correcting capabilities. The result is an optimal worst-case QoS routing algorithm that uses the Generalized Dijkstra's Algorithm as a subroutine with a polynomial time complexity of O(V3).

The performance of sphere decoding of block codes over a variety of channels is investigated. We derive a tight bound on the performance of maximum likelihood decoding of linear codes on q-ary symmetric channels. We use this result to bound the performance of q-ary hard decision sphere decoders. We also derive a tight bound on the performance of soft decision sphere decoders on the AWGN channel for BPSK and M-PSK modulated block codes. The performance of soft decision sphere decoding of arbitrary finite lattices or block codes is also analyzed

By definition Huffman codes only exist for finite sources, since the Huffman algorithm cannot be applied to an infinite source. On the other hand, Gallager's sibling property, which was introduced as a characterization of Huffman codes, extends naturally to (countably) infinite sources. Thus we define a Huffman-Gallager code as any code that has the sibling property, and we present some basic facts about such codes. (1) For any source, a Huffman-Gallager code exists and its list of node probabilities is unique. (2) A Huffman-Gallager code is optimal, and given an optimal code, there exists a Huffman-Gallager code with the same codeword lengths. (3) For sources with infinite entropy, the existence and uniqueness results continue to hold, and the optimality results hold for a natural extended form of optimality

In AG coding theory is very important to work with curves with many rational points, to get good codes. In this paper, from curves defined over F2with genus g ges 1 we give sufficient conditions for getting maximal curves over F2E2g

In this paper, we present an iterative soft-decision decoding algorithm for Reed-Solomon (RS) codes offering both complexity and performance advantages over previously known decoding algorithms. Our algorithm is a list decoding algorithm which combines two powerful soft-decision decoding techniques which were previously regarded in the literature as competitive, namely, the Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation based on adaptive parity-check matrices, recently proposed by Jiang and Narayanan. Building on the Jiang-Narayanan algorithm, we present a belief-propagation-based algorithm with a significant reduction in computational complexity. We introduce the concept of using a belief-propagation-based decoder to enhance the soft-input information prior to decoding with an algebraic soft-decision decoder. Our algorithm can also be viewed as an interpolation multiplicity assignment scheme for algebraic soft-decision decoding of RS codes.

A closed form formula of the partition weight enumerator of maximum distance separable (MDS) codes is derived for an arbitrary number of partitions. Using this result, some properties of MDS codes are discussed. The results are extended for the average binary image of MDS codes in finite fields of characteristic two. As an application, we study the multiuser error probability of Reed Solomon codes

This paper considers the problem of finding average enumerators for the class of protograph ensembles, which are related in a certain way to quasi-cyclic codes. Our methods, which are necessarily different from those used to compute enumerators for classical irregular ensembles, can be applied to both codeword and stopping set weight enumerators. The method divides codewords into types based on their partial weight enumerator. For each type, an exponent can be computed for the average number of codewords of that type. Maximizing over types of fixed average weight gives the average enumerator which we seek. Although this maximization step is in general difficult because of non-unique local maxima, we can compute it for simple cases. We show that certain ensembles exist which have a linearly growing minimum distance with high probability, while others have at most sublinearly growing minimum distance with high probability

We present a method for designing high-rate, high-performance SCTCM (serially concatenated trellis coded modulation) systems with in-line interleavers. Using in-line EXIT charts and ML performance analysis, we develop criteria for choosing constituent codes and optimization methods for selecting the best ones. To illustrate our methods, we show that an optimized SCTCM system with an in-line interleaver for rate r = 5/6 and 64QAM has better performance than other turbo-like TCMs with the same parameters.

In an attempt to determine the ultimate capabilities of the Sudan-Guruswami-Sudan-Kotter-Vardy algebraic soft decision decoding algorithm for Reed-Solomon codes, we present a new method, based on the Chernoff bound, for constructing multiplicity matrices. In many cases, this technique predicts that the potential performance of ASD decoding of RS codes is significantly better than previously thought.

This paper presents the data collection function in sensory networks. Specifically we derive relationships between data collection time and transmission range, data packet size and channel noise in the simple line scenario. To develop intuition these relationships are studied in the limit case where the number of sensor nodes becomes large.

We show the capacity of a generalized pulse position modulation (PPM) channel, where the input vectors may be any set that allows a transitive group of coordinate permutations, is achieved by a uniform input distribution. We derive a simple expression for capacity in terms of the Kullback-Leibler distance for the binary case, and find the asymptote in the PPM order. We prove a subadditivity result for the PPM channel and use it to show PPM capacity is monotonic in the order.

Data collection, i.e., the aggregation at the user location of information gathered by sensor nodes, is a fundamental function of sensory networks. Indeed, most sensor network applications rely on data collection capabilities, and consequently, an inefficient data collection process may adversely affect the performance of the network. In this paper, we study via simple discrete mathematical models, the time performance of the data collection and data distribution tasks in sensory networks. Specifically, we derive the minimum delay in collecting sensor data for networks of various topologies such as line, multiline, and tree and give corresponding optimal scheduling strategies. Furthermore, we bound the data collection time on general graph networks. Our analyses apply to networks equipped with directional or omnidirectional antennas and simple comparative results of the two systems are presented.

In this paper, we study, via simple discrete mathematical models, the problems of data distribution and data collection in wireless sensor networks. The work that follows continues the work presented by the authors in (C. Florens et al., 2002) where the focus was on sensor networks equipped with unidirectional antenna elements. Here we shift our interest to networks equipped with omnidirectional antenna elements. In particular we show how the data distribution and collection tasks can be performed optimally (with respect to time) on tree networks and give the corresponding time performances of those strategies. We also present a strategy for general graph networks that performs within a factor of 3 of the optimal performance. Finally we compare the performance of a network equipped with omnidirectional antenna elements with one equipped with unidirectional antenna elements. We show the latter outperforms the former by 33% at most in tree networks. To that purpose we included relevant results on directional antenna sensor networks, partly obtained in (C.Florens et al., 2002).

We define a simple discrete mathematical model for wireless ad-hoc sensor networks and study the problems of data distribution and data collection which arise in those networks. We show how those tasks can be optimally performed on tree sensor networks that use directional antennas. Furthermore we compare the performance of a network equipped with directional antenna elements with one equipped with omnidirectional antenna elements and show the former outperforms the latter by 50% at most on a line network.

This paper is devoted to a Shannon-theoretic study of turbo codes. We prove that ensembles of parallel and serial turbo codes are "good" in the following sense. For a turbo code ensemble defined by a fixed set of component codes (subject only to mild necessary restrictions), there exists a positive number /spl gamma//sub 0/ such that for any binary-input memoryless channel whose Bhattacharyya noise parameter is less than /spl gamma//sub 0/, the average maximum-likelihood (ML) decoder block error probability approaches zero, at least as fast as n/sup -/spl beta//, where /spl beta/ is the "interleaver gain" exponent defined by Benedetto et al. in 1996.

In the 50 years since Shannon determined the capacity of ergodic channels, the construction of capacity-approaching coding schemes has been the supreme goal of coding research. Finally today, we know of practical codes and decoding algorithms that can closely approach the channel capacity of some classical memoryless channels. It is a remarkable fact motivating this special issue that all known practical, capacity-approaching coding schemes are now understood to be codes defined on graphs, together with the associated iterative decoding algorithms.

We give a proof of a theorem on the common divisibility of polynomials and permuted polynomials (over GF(2)) by a polynomial g(x).

We discuss a general message passing algorithm, which we call the generalized distributive law (GDL). The GDL is a synthesis of the work of many authors in information theory, digital communications, signal processing, statistics, and artificial intelligence. It includes as special cases the Baum-Welch algorithm, the fast Fourier transform (FFT) on any finite Abelian group, the Gallager-Tanner-Wiberg decoding algorithm, Viterbi's algorithm, the BCJR algorithm, Pearl's "belief propagation" algorithm, the Shafer-Shenoy probability propagation algorithm, and the turbo decoding algorithm. Although this algorithm is guaranteed to give exact answers only in certain cases (the "junction tree" condition), unfortunately not including the cases of GTW with cycles or turbo decoding, there is much experimental evidence, and a few theorems, suggesting that it often works approximately even when it is not supposed to.

We describe the close connection between the now celebrated iterative turbo decoding algorithm of Berrou et al. (1993) and an algorithm that has been well known in the artificial intelligence community for a decade, but which is relatively unknown to information theorists: Pearl's (1982) belief propagation algorithm. We see that if Pearl's algorithm is applied to the "belief network" of a parallel concatenation of two or more codes, the turbo decoding algorithm immediately results. Unfortunately, however, this belief diagram has loops, and Pearl only proved that his algorithm works when there are no loops, so an explanation of the experimental performance of turbo decoding is still lacking. However, we also show that Pearl's algorithm can be used to routinely derive previously known iterative, but suboptimal, decoding algorithms for a number of other error-control systems, including Gallager's (1962) low-density parity-check codes, serially concatenated codes, and product codes. Thus, belief propagation provides a very attractive general methodology for devising low-complexity iterative decoding algorithms for hybrid coded systems.

We introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed /spl nu/-dimensional vector subspace S of GF (2/sup m/). SSRS codes are constructed using properties of the Galois field GF(2/sup m/). They are not linear over the field GF(2/sup /spl nu//), which does not come into play, but rather are Abelian group codes over S. However, they are linear over GF(2), and the symbol-wise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that (1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF(2/sup /spl nu//) is a subfield of GF(2/sup m/), it may not be the best /spl nu/-dimensional subspace for constructing SSRS codes; and (2) many high-rate SSRS codes have a larger dimension than any previously known code with the same values of n, d, and q, including algebraic-geometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems.

In this partially tutorial paper, we examine minimal trellis representations of linear block codes and analyze several measures of trellis complexity: maximum state and edge dimensions, total span length, and total vertices, edges and mergers. We obtain bounds on these complexities as extensions of well-known dimension/length profile (DLP) bounds. Codes meeting these bounds minimize all the complexity measures simultaneously; conversely, a code attaining the bound for total span length, vertices, or edges, must likewise attain it for all the others. We define a notion of "uniform" optimality that embraces different domains of optimization, such as different permutations of a code or different codes with the same parameters, and we give examples of uniformly optimal codes and permutations. We also give some conditions that identify certain cases when no code or permutation can meet the bounds. In addition to DLP-based bounds, we derive new inequalities relating one complexity measure to another, which can be used in conjunction with known bounds on one measure to imply bounds on the others. As an application, we infer new bounds on maximum state and edge complexity and on total vertices and edges from bounds on span lengths.

In this semi-tutorial paper, we will investigate the computational complexity of an abstract version of the Viterbi algorithm on a trellis, and show that if the trellis has e edges, the complexity of the Viterbi algorithm is /spl Theta/(e). This result suggests that the "best" trellis representation for a given linear block code is the one with the fewest edges. We will then show that, among all trellises that represent a given code, the original trellis introduced by Bahl, Cocke, Jelinek, and Raviv in 1974, and later rediscovered by Wolf (1978), Massey (1978), and Forney (1988), uniquely minimizes the edge count, as well as several other figures of merit. Following Forney and Kschischang and Sorokine (1995), we will also discuss "trellis-oriented" or "minimal-span" generator matrices, which facilitate the calculation of the size of the BCJR trellis, as well as the actual construction of it.

Convolutional codes have a natural, regular, trellis structure that facilitates the implementation of Viterbi's algorithm. Linear block codes also have a natural, though not in general a regular, "minimal" trellis structure, which allows them to be decoded with a Viterbi-like algorithm. In both cases, the complexity of an unenhanced Viterbi decoding algorithm can be accurately estimated by the number of trellis edge symbols per encoded bit. It would therefore appear that we are in a good position to make a fair comparison of the Viterbi decoding complexity of block and convolutional codes. Unfortunately, however, this comparison is somewhat muddled by the fact that some convolutional codes, the punctured convolutional codes, are known to have trellis representations which are significantly less complex than the conventional trellis. In other words, the conventional trellis representation for a convolutional code may not be the "minimal" trellis representation. Thus ironically, we seem to know more about the minimal trellis representation for block than for convolutional codes. We provide a remedy, by developing a theory of minimal trellises for convolutional codes. This allows us to make a direct performance-complexity comparison for block and convolutional codes. A by-product of our work is an algorithm for choosing, from among all generator matrices for a given convolutional code, what we call a trellis-canonical generator matrix, from which the minimal trellis for the code can be directly constructed. Another by-product is that in the new theory, punctured convolutional codes no longer appear as a special class, but simply as high-rate convolutional codes whose trellis complexity is unexpectedly small.

Studies the performance of channel assignment algorithms for "channelized" (e.g., FDMA or TDMA) cellular telephone systems, via mathematical models, each of which is characterized by a pair (H,p), where H is a hypergraph describing the channel reuse restrictions, and p is a probability vector describing the variation of traffic intensity from cell to cell. For a given channel assignment algorithm, the authors define T(r) to be the amount of carried traffic, as a function of the offered traffic, where both r and T(r) are measured in Erlangs per channel. They show that for a given H and p, there exists a function T/sub H,p/(r), which can be computed by linear programming, such that for every channel assignment algorithm, T(r)/spl les/T/sub H,p/(r). Moreover, they show that there exist channel assignment algorithms whose performance approaches T/sub H,p/(r) arbitrarily closely as the number of channels increases. As a corollary, they show that for a given (H,p) there is a number r/sub 0/, which also can be computed by linear programming, such that if the offered traffic exceeds r/sub 0/, then for any channel assignment algorithm, a positive fraction of all call requests must be blocked, whereas if the offered traffic is less than r/sub 0/, all call requests can be honored, if the number of channels is sufficiently large. The authors call r/sub 0/, whose units are Erlangs per channel, the capacity of the cellular system.>

Various aspects of single-phased burst-error-correcting array codes are explored. These codes are composed of two-dimensional arrays with row and column parities with a diagonally cyclic readout order; they are capable of correcting a single burst error along one diagonal. Optimal codeword sizes are found to have dimensions n/sub 1/*n/sub 2/ such that n/sub 2/ is the smallest prime number larger than n/sub 1/. These codes are capable of reaching the Singleton bound. A new type of error, approximate errors, is defined; in q-ary applications, these errors cause data to be slightly corrupted and therefore still close to the true data level. Phased burst array codes can be tailored to correct these codes with even higher rates than before.>

The authors present a study of the performance of a general class of channel assignment algorithms. These algorithms, which they call Omega -algorithms, are completely characterized by the set of carried-traffic "states" which they allow. They show that for any such algorithm, there is a closed-form expression for the carried traffic function, which lends itself to several kinds of asymptotic analysis. As an application, they study a particular Omega -algorithm, which has been previously studied under the name "maximum packing algorithm", and which is a "greedy" dynamic channel assignment algorithm, and show that its performance is in many cases inferior to that of simple fixed channel assignment algorithms. They show that the cause of this unexpected phenomenon is the tendency of dynamic algorithms to get trapped in states that are locally, but not globally, maximal.>

The deBruijn graph B n is the state diagram for an n -stage binary shift register. It has 2 n vertices and 2 n + 1 edges. In this papers, it is shown that B n can be built by appropriately “wiring together“ (i.e., connecting together with extra edges) many isomorphic copies of a fixed graph, which is called a building block for B n . The efficiency of such a building block is refined as the fraction of the edges of B n which are present in the copies of the building block. It is then shown, among other things, that for any α < 1, there exists a graph G which is a building block for B n of efficiency > α for all sufficiently large n . These results are illustrated by describing how a special hierarchical family of building blocks has been used to construct a very large Viterbi decoder (whose floorplan is the graph B 13 ) which will be used on NASA's Galileo mission.

The performance of general binary block codes on an unquantized additive white Gaussian noise (AWGN) channel at low signal-to-noise ratios is considered. Expressions are derived for both the block error and the bit error probabilities near the point where the bit signal-to-noise ratio is zero. These expressions depend on the global geometric structure of the code, although the minimum distance still seems to play a crucial role. Examples of codes such as orthogonal codes, biorthogonal codes, the (24,12) extended Golay code, and the (15,6) expurgated BCH code are discussed. The asymptotic coding gain at low signal-to-noise ratios is also studied.>

The lifetimes of computer memories which are protected with single-error-correcting-double-error-detecting (SEC-DED) codes are studies. The authors assume that there are five possible types of memory chip failure (single-cell, row, column, row-column and whole chip), and, after making a simplifying assumption (the Poisson assumption), have substantiated that experimentally. A simple closed-form expression is derived for the system reliability function. Using this formula and chip reliability data taken from published tables, it is possible to compute the mean time to failure for realistic memory systems.>

A new class of codes, called burst identification codes, is defined and studied. These codes can be used to determine the patterns of burst errors. Two-dimensional burst correcting codes can be easily constructed from burst identification codes. The resulting class of codes is simple to implement and has lower redundancy than other comparable codes. The results are pertinent to the study of radiation effects on VLSI RAM chips, which can cause two-dimensional bursts of errors.>

Huffman's algorithm gives optimal codes, as measured by average codeword length, and the redundancy can be measured as the difference between the average codeword length and Shannon's entropy. If the objective function is replaced by an exponentially weighted average, then a simple modification of Huffman's algorithm gives optimal codes. The redundancy can now be measured as the difference between this new average and A. Renyi's (1961) generalization of Shannon's entropy. By decreasing some of the codeword lengths in a Shannon code, the upper bound on the redundancy given in the standard proof of the noiseless source coding theorem is improved. The lower bound is improved by randomizing between codeword lengths, allowing linear programming techniques to be used on an integer programming problem. These bounds are shown to be asymptotically equal. The results are generalized to the Renyi case and are related to R.G. Gallager's (1978) bound on the redundancy of Huffman codes.>

A class of codes called finite-state (FS) codes is defined and investigated. The codes, which generalize both block and convolutional codes, are defined by their encoders, which are finite-state machines with parallel inputs and outputs. A family of upper bounds on the free distance of a given FS code is derived. A general construction for FS codes is given, and it is shown that in many cases the FS codes constructed in this way have a free distance that is the largest possible. Catastrophic error propagation (CEP) for FS codes is also discussed. It is found that to avoid CEP one must solve the graph-theoretic problem of finding a uniquely decodable edge labeling of the state diagram.>

The capacity of channels with block memory is investigated. It is shown that, when the problem is modeled as a game-theoretic problem, the optimum coding and noise distributions when block memory is permitted are independent from symbol to symbol within a block. Optimal jamming strategies are also independent from symbol to symbol within a block.>

Recently, Posner noted that on a wide-band Gaussian broadcast channel, ordinary time-shared coding performs almost as well as more sophisticated broadcast coding strategies. In this note, we shall give a quantitative version of Posner's result and argue that for certain realistic broadcast channels time sharing may suffice.

Techniques from coding theory are applied to study rigorously the capacity of the Hopfield associative memory. Such a memory storesn-tuple of\pm 1's. The components change depending on a hard-limited version of linear functions of all other components. With symmetric connections between components, a stable state is ultimately reached. By building up the connection matrix as a sum-of-outer products ofmfundamental memories, one hopes to be able to recover a certain one of themmemories by using an initialn-tuple probe vector less than a Hamming distancen/2away from the fundamental memory. Ifmfundamental memories are chosen at random, the maximum asympotic value ofmin order that most of themoriginal memories are exactly recoverable isn/(2 \log n). With the added restriction that every one of themfundamental memories be recoverable exactly,mcan be no more thann/(4 \log n)asymptotically asnapproaches infinity. Extensions are also considered, in particular to capacity under quantization of the outer-product connection matrix. This quantized memory capacity problem is closely related to the capacity of the quantized Gaussian channel.

It is shown that for each integerb \geq 1infinitely many optimum cyclicb-burst-correcting codes exist, i.e., codes whose lengthn, redundancyr, and burst-correcting capabilityb, satisfyn = 2^{r-b+1} - 1. Some optimum codes forb = 3, 4, and5are also studied in detail.

Motivated by a problem in optical communication, we consider the general problem of maximizing the entropy of a stationary random process that is subject to an average transition cost constraint. Using a recent result of Justesen and Hoholdt, we present an exact solution to the problem and suggest a class of finite state encoders that give a good approximation to the exact solution.

Upper bounds On the decoder error probability for Reed-Solomon codes are derived. By definition, "decoder error" occurs when the decoder finds a codeword other than the transitted codeword; this is in contrast to "decoder failure," which occurs when the decoder fails to find any codeword at all. These results imply, for example, that for aterror-correcting Reed-Solomon code of lengthq - 1over GF(q), if more thanterrors occur, the probability of decoder error is less than1/t!.

Patel and Hong have constructed a code that can correct any track error or two track erasures in a9-track magnetic tape. Here the construction is extended to a code that can correct a track error and a track erasure or three track erasures. A generalization is given.

Motivated by the needs of NASA's Voyager 2 mission, in this paper we describe an algorithm which detects and corrects losses of node synchronization in convolutionally encoded data. This algorithm, which would be implemented as a hardware device external to a Viterbi decoder, makes statistical decisions about node synch based on the hard-quantized undecoded data stream. We will show that in a worst-case Voyager environment, our method will detect and correct a true loss of synch (thought to be a very rare event) within several hundred bits; many of the resulting outages will be corrected by the outer Reed-Solomon code. At the same time, the mean time between false alarms is on the order of several years, independent of the signal-to-noise ratio.

A new class of channel models with memory is presented in order to study various kinds of interference phenomena. It is shown, among other things, that when all other parameters are held fixed, channel capacityCis an {\em increasing} function of the memory length, while the cutoff rateR_{0}generally is a {\em decreasing} function. Calculations with various explicit coding schemes indicate thatCis better thanR_{0}as a performance measure for these channel models. As a partial resolution of thisCversusR_{0}paradox, the conjecture is offered thatR_{0}is more properly a measure of coding delay rather than of coding complexity.

The capacity region for the discrete memoryless multiple-access channel without time synchronization at the transmitters and receivers is shown to be the same as the known capacity region for the ordinary multiple-access channel. The proof utilizes time sharing of two optimal codes for the ordinary multiple-access channel and uses maximum likelihood decoding over shifts of the hypothesized transmitter words.

Although the existence of universal noiseless variable-rate codes for the class of discrete stationary ergodic sources has previously been established, very few practical universal encoding methods are available. Efficient implementable universal source coding techniques are discussed in this paper. Results are presented on source codes for which a small value of the maximum redundancy is achieved with a relatively short block length. A constructive proof of the existence of universal noiseless codes for discrete stationary sources is first presented. The proof is shown to provide a method for obtaining efficient universal noiseless variable-rate codes for various classes of sources. For memoryless sources, upper and lower bounds are obtained for the minimax redundancy as a function of the block length of the code. Several techniques for constructing universal noiseless source codes for memoryless sources are presented and their redundancies are compared with the bounds. Consideration is given to possible applications to data compression for certain nonstationary sources.

In a recent paper, Pierce studied the problems of communicating at optical frequencies using photon-counting techniques, and concluded that "at low temperatures we encounter insuperable problems of encoding long before we approach [channel capacity]." In this paper it is shown that even assuming a noiseless model for photon communication for which capacity (measured in nats/photon) is infinite, it is unlikely that a signaling efficiency of even 10 nats/photon could be achieved practically. On the positive side, it is shown that pulse-position modulation plus Reed-Solomon coding yields practical results in the range of 2 to 3 nats/photon.

Alternate symbol inversion is sometimes applied to the output of convolutional encoders to guarantee sufficient richness of symbol transition for the receiver symbol synchronizer. A bound is given for the length of the transition-free symbol stream in such systems, and those convolutional codes are characterized in which arbitrarily long transition free runs occur.

The fact that the general decoding problem for linear codes and the general problem of finding the weights of a linear code are both NP-complete is shown. This strongly suggests, but does not rigorously imply, that no algorithm for either of these problems which runs in polynomial time exists.

The recent upper bounds on the minimum distance of binary codes given by McEliece, Rodemich, Rumsey, and Welch are shown to result in improved upper bounds on the block coding error exponent for binary-input memoryless channels.

With the Delsarte-MacWilliams inequalities as a starting point, an upper bound is obtained on the rate of a binary code as a function of its minimum distance. This upper bound is asymptotically less than Levenshtein's bound, and so also Elias's.

An example is provided of two convolutional codes that have the same transmission gain but whose dual codes do not. This shows that no analog of the MacWilliams identity for block codes can exist relating the transmission gains of a convolutional code and its dual.

An upper bound on the minimum distance of binary blocks codes, which is superior to Elias' bound for R < 0.0509^+, is obtained. The new bound has the same derivative(-infty) at R = 0 as Gilbert's lower bound. (Elias' bound has derivative-ln 2 at R = 0).

Griesmer's lower bound for the word length n of a linear code of dimension k and minimum distance d is shown to be sharp for fixed k, when d is sufficiently large. For k ≤ 6 and all d the minimum word length is determined.

In the above paper [1], Varshamov considers discrete channels with q inputs and q outputs, q being an arbitrary integer.

It is shown that there are arbitrarily long "good" (in the sense of Gilbert) binary block codes that are preserved under very large permutation groups. This result contrasts sharply with the properties of linear codes: it is conjectured that long cyclic codes are bad, and known that long affine-invariant codes are bad.