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Much has been made (especially in terms of royalties) of the supposed codes to be found in the works of Leonardo da Vinci, and their relation to a series of [...]

In this correspondence, we present an interleaving scheme that yields quasi-cyclic turbo codes. We prove that randomly chosen members of this family yield with probability almost 1 turbo codes with asymptotically optimum minimum distance, i.e., growing as a logarithm of the interleaver size. These interleavers are also very practical in terms of memory requirements and their decoding error probabilities for small block lengths compare favorably with previous interleaving schemes. Index Terms--Convolutional codes, iterative decoding, minimum distance, quasi-cyclic codes, turbo codes.

In this correspondence, we introduce a new class of low-rate error correction codes called zigzag Hadamard (ZH) codes and their concatenation schemes. Each member of this class of codes is specified by a highly structured zigzag graph with each segment being a Hadamard codeword. The ZH codes enjoy extremely simple encoding and very low-complexity soft-input-soft-output (SISO) decoding based on a posteriori probability (APP) fast Hadamard transform (FHT) technique. We present an asymptotic performance analysis of the proposed concatenated ZH codes using the extrinsic mutual information transfer (EXIT) chart for infinite-length codes. We also provide a union bound analysis of the error performance for finite-length codes. Furthermore, the concatenated ZH codes are shown to be a good class of codes in the low-rate region. Specifically, a rate-0.0107 concatenated code with three ZH components and an interleaver size of 65536 can achieve the bit error rate (BER) performance of [10.-sup.-5] at -1.15 dB, which is only 0.44 dB away from the ultimate Shannon limit. The proposed concatenated ZH codes offer similar performance as another class of low-rate codes-the turbo-Hadamard codes, and better performance than superorthogonal turbo codes, with much lower encoding and decoding complexities. Index Terms--Extrinsic mutual information transfer (EXIT) chart, Hadamard codes, low-complexity decoding, low rate, parallel concatenation, union bound, zigzag codes.

In this letter, we present an analysis tool for the convergence of a turbo decoder using the min-sum algorithm. First of all, we classify cycles on a turbo code factor graph. We define and develop a quantity which characterizes the trellis convergence and a probability of a message round on a cycle. Our analysis is then applied to an interleaver design for a turbo decoder using the min-sum algorithm. Since the probability of a message round is closely related to the convergence property, the constructed interleaver optimizes the convergence property. The performance is compared with several referential interleavers. Index Terms--Convergence, interleaver design, iterative decoding, message propagation, turbo codes.

We generalize a characterization of p-ary (p is a prime) quantum codes given by Feng and Xing to q-ary (q is a prime power) quantum codes. This characterization makes it possible to convert an asymptotic bound of Stichtenoth and Xing for nonlinear algebraic geometry codes to a quantum asymptotic bound. Besides, we also investigate the asymptotic behavior of quantum codes. Index Terms--Algebraic-geometry codes, asymptotic behavior, propagation rules.

A code developed for calculating dose rate maps in scenarios with complex geometries and gamma radiation sources is presented in this work. It is based on the point kernel method. The code computes photon flux and dose rate in preselected positions due to volumetric radioactive sources in the presence of other nonemitting materials acting as shielding. Buildup factors are considered to account for radiation scattering effects, using the Geometric-Progression formula in the fitting function. The major advantage of this code is that there are no constraints for the generation of geometries. Any type and number of volumes can be loaded to generate the scenario by using standard three-dimensional computer-aided design tools. The geometry is imported as a DXF format file. In a second step, material and radiometric attributes are imported for each volume. Volumetric radioactive sources are represented by a set of single emitting point sources distributed within its geometry. The code output has been verified in three different ways: by comparing with results from other point-kernel packages, by comparing with estimates obtained with a Monte Carlo code, and by contrasting with direct experimental measurements. Results of these comparisons are presented and discussed. Finally, intrinsic limitations of the code are discussed. Index Terms--Dose rate calculation, point kernel, three-dimensional (3-D) modeling.

We investigate the application of low-density parity-check (LDPC) codes in volume holographic memory (VHM) systems. We show that a carefully designed irregular LDPC code has a very good performance in VHM systems. We optimize high-rate LDPC codes for the nonuniform error pattern in holographic memories to reduce the bit error rate extensively. The prior knowledge of noise distribution is used for designing as well as decoding the LDPC codes. We show that these codes have a superior performance to that of Reed-Solomon (RS) codes and regular LDPC counterparts. Our simulation shows that we can increase the maximum storage capacity of holographic memories by more than 50 percent if we ,use irregular LDPC codes with soft-decision decoding instead of conventionally employed RS codes with hard-decision decoding. The performance of these LDPC codes is close to the information theoretic capacity. OCIS codes: 210.2860, 200.3050.

Hybrid In Band on Channel (IBOC) digital audio broadcasting simultaneously with analog amplitude modulation (AM) has been proposed as a hybrid solution to digital audio broadcasting in the AM band. Since the AM band is crowded and since the available bandwidth per program is limited, adding digital transmission is a challenging proposition. To achieve FM like audio quality, an audio coder rate of 32-64 kb/sec may be required. One of the currently proposed hybrid IBOC-AM systems is 30 kHz wide. Severe second adjacent interference may occur in certain geographical areas. This may lead to loss of 40% of the effective transmission audio bit rate. For coping with such harsh transmission conditions, we present a solution based on embedded/multidescriptive audio coding with matched multistream transmission in separate frequency bands. With loss of one frequency band, the embedded system blends to a lower audio coder rate with a much better quality than analog AM. The nonembedded system without multistream transmission fails catastrophically when a little more than one sideband is severely interfered with causing a severe discontinuity in quality while blending directly to analog AM. A number of detailed robust embedded systems are outlined. We also show how multistream transmission schemes can be used with nonembedded audio coders. Both daytime and nighttime scenarios are included. This paper contains a catalog of possible systems for different audio quality levels and interference scenarios, including systems with 20 kHz bandwidth rather than 30 kHz. Index Terms--Amplitude modulation, audio coding, channel coding, digital audio broadcasting, error detection coding, radio broadcasting.

The rank and kernel of 1-perfect additive codes is determined. Additive codes could be seen as translation-invariant propelinear codes and, in this correspondence, a characterization of propelinear codes as codes having a regular subgroup of the full group of isometries of the code is established. A characterization of the automorphism group of a 1-perfect additive code is given and also the cardinality of this group is computed. Finally, an efficiently computable characterization of the Steiner triple systems associated with a 1-perfect binary additive code is also established. Index Terms--Additive codes, automorphism group, kernel, 1-perfect codes, rank.

Bounds on the redundancy of the recency-rank and block-sorting universal lossless data compression algorithms for finite-length sequences are presented. These algorithms are asymptotically optimal for infinite-length sequences, stationary ergodic sources in the almost-sure sense, and asymptotically mean stationary sources in the average and almost-sure sense. Index Terms--Block-sorting algorithm, book-stack algorithm, Burrows-Wheeler transform, lossless data compression, move-to-front algorithm, recency-rank algorithm, universal code.

We consider the search problem in which one finds a binary word among m words chosen randomly from the set of all words of fixed length n. It is well known that the optimal search is equivalent to the Huffman coding that requires on average [log.sub.2] m bits to be checked plus a small additional cost called the average redundancy. The latter is an oscillating function of m and is bounded between zero and 1 - (1 + ln ln 2)/ln 2 [approximately equal to] 0.0860713320. As a matter of fact, it is known that finding the optimal strategy for this problem is NP-hard. We propose here several simple randomized search strategies leading, respectively, to the following average redundancies: 1.332746177, 0.6113986565, 0.4310617764, and 0.332746177, plus some small oscillations that we precisely characterize. These results should be compared to the optimal, but NP-hard, search algorithm. Our findings extend and make more precise recent results of Fedotov and Ryabko. Index Terms--Generating functions, Huffman code, PATRICIA trie, search problem, Shannon source coding theorem, tries.

This correspondence shows the equivalence of three previously proposed reduced-rank detection schemes for direct-sequence code-division multiple-access (DS-CDMA) communication systems. The auxiliary vector filtering (AVF) algorithm is simplified through a key observation on the construction of the auxiliary vectors. After simplification, it is shown that the AVF algorithm is equivalent to the multistage Wiener filtering (MWF) algorithm of Honig and Goldstein. Furthermore, these schemes can be shown to be equivalent to the multistage linear receiver scheme based on the Cayley-Hamilton (CH) theorem when the minimum mean-square error (MMSE) criterion is applied to the reduced dimensional space of the received signal.

We characterize codes over [Z.sub.p] which are the Gray images of (1 - [p.sup.k])-cyclic codes or cyclic codes over [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (k [greater than or equal to] 1). A necessary and sufficient condition for the Gray image of a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-linear (1 - p)-cyclic code to be linear is given. In many cases, this yields an explicit description of the Gray image of a linear (1 - p)-cyclic code over [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], of length relatively prime to p. Linear cyclic codes over [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a whose Gray images are linear cyclic codes over [Z.sub.p] have been characterized. Some generalizations of these results to the case of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where k [greater than or equal to] 2, are also obtained. Index Terms--Constacyclic code, cyclic code, Gray map, linear code, quasi-cyclic code.

The slope of the active distances is an important parameter when investigating the error-correcting capability of convolutional codes and the distance behavior of concatenated convolutional codes. The slope of the active distances is equal to the minimum average weight cycle in the state-transition diagram of the encoder. A general upper bound on the slope depending on the free distance of the convolutional code and new upper bounds on the slope of special classes of binary convolutional codes are derived. Moreover, a search technique, resulting in new tables of rate R = 1/2 and rate R = 1/3 convolutional encoders with high memories and large active distance-slopes is presented. Furthermore, we show that convolutional codes with large slopes can be used to obtain new tailbiting block codes with large minimum distances. Tables of rate R = 1/2 and rate R = 1/3 tailbiting codes with larger minimum distances than the best previously known quasi-cyclic codes are given. Two new tailbiting codes also have larger minimum distances than the best previously known binary linear block codes with same size and length. One of them is also superior in terms of minimum distance to any previously known binary nonlinear block code with the same set of parameters. Index Terms--Active distances, convolutional codes, minimum distance, quasi-cyclic codes, tailbiting codes.

We derive upper bounds on the rate of low-density parity-check (LDPC) codes for which reliable communication is achievable. We first generalize Gallager's bound to a general binary-input symmetric-output channel. We then proceed to derive tighter bounds. We also derive upper bounds on the rate as a function of the minimum distance of the code. We consider both individual codes and ensembles of codes. Index Terms--Error probability, iterative decoding, low-density parity-check (LDPC) codes, maximum-likelihood (ML) decoding, minimum distance.

This paper focuses on jammed timing channels. Pure delay jammers with a maximum delay constraint, an average delay constraint, or a maximum buffer size constraint are explored, for continuous-time or discrete-time packet waveforms. Fluid waveform approximations of each of these classes of waveforms are employed to aid in analysis. Channel capacity is defined and an information-theoretic game based on mutual information rate is studied. Min-max optimal jammers and max-min optimal input processes are sought. Bounds on the min-max and max-min mutual information rates are described, and numerical examples are given. For maximum-delay-constrained (MDC) jammers with continuous-time packet waveforms, saddle-point input and jammer strategies are identified. The capacity of the maximum-delay constrained jamming channel with continuous-time packet waveforms is shown to equal the mutual information rate of the saddle point. For MDC jammers with discrete-time packet waveforms, saddle-point strategies are shown to exist. Jammers which have quantized batch departures at regular intervals are shown to perform well. Input processes with batches at regular intervals perform well for MDC or maximum-buffer-size-constrained jammers. Index Terms--Channel coding, covert timing channels, jamming, network security.

Recently, (linear) codes over [Z.sub.4] and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study [Z.sub.4]-QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to [Z.sub.4] produces a new binary code, a (92, [2.sub.24], 28)-code, which is the best among all binary codes (linear or nonlinear). Moreover, we characterize cyclic codes corresponding to free modules in terms of their generator polynomials. Index Terms--Gray map, quasi-cyclic (QC) codes, [Z.sub.4] codes.

Binary prefix-free codes in which all codewords end with a "1" have been introduced by Berger and Yeung. A recursive method is given here for the construction of all optimal "1"-ended codes with n codewords. It is shown that the set of codes obtained by the construction contains only optimal codes. We also compute recursively the number of essentially different optimal "1"-ended codes with n codewords and show that their number grows faster than any polynomial in n. Index Terms--One-ended codes, optimal codes, prefix-free codes, variable-length coding.

In this correspondence, we evaluate the variance of the union performance bound for a rate-1/3 turbo code over all possible interleavers of length N, under the assumption of a maximum-likelihood (ML) decoder. Theoretical and simulation results for turbo codes with two-memory component codes indicate that the coefficient of variation of the bound increases with the signal-to-noise ratio and decreases with the interleaver length. Theoretical analysis for large interleaver lengths shows that the coefficient of variation asymptotically approaches a constant value. The results also demonstrate that the majority of the interleavers have performance bounds very close to the average value of the bound. This phenomenon is more palpable for larger interleaver lengths. Index Terms--Channel coding, concatenated codes, turbo codes, union performance bound.

In this correspondence, new extremal self-dual codes of length 62 are constructed with weight enumerators of three different types. Two of these types were not represented by any known code up till now. All these codes possess an automorphism of order 15. Some of them are used to construct extremal self-dual codes of length 60 by the method of subtracting.

Multiple-antenna systems that operate at high rates require simple yet effective space-time transmission schemes to handle the large traffic volume in real time. At rates of tens of bits per second per hertz, Vertical Bell Labs Layered Space-Time (V-BLAST), where every antenna transmits its own independent substream of data, has been shown to have good performance and simple encoding and decoding. Yet V-BLAST suffers from its inability to work with fewer receive antennas than transmit antennas--this deficiency is especially important for modern cellular systems, where a base station typically has more antennas than the mobile handsets. Furthermore, because V-BLAST transmits independent data streams on its antennas there is no built-in spatial coding to guard against deep fades from any given transmit antenna. On the other hand, there are many previously proposed space-time codes that have good fading resistance and simple decoding, but these codes generally have poor performance at high data rates or with many antennas. We propose a high-rate coding scheme that can handle any configuration of transmit and receive antennas and that subsumes both V-BLAST and many proposed space-time block codes as special cases. The scheme transmits substreams of data in linear combinations over space and time. The codes are designed to optimize the mutual information between the transmitted and received signals. Because of their linear structure, the codes retain the decoding simplicity of V-BLAST, and because of their information-theoretic optimality, they possess many coding advantages. We give examples of the codes and show that their performance is generally superior to earlier proposed methods over a wide range of rates and signal-to-noise ratios (SNRs). Index Terms--Bell Labs Layered Space-Time (BLAST), fading channels, multiple antennas, receive diversity, space-time codes, transmit diversity, wireless communications.

In this paper, we characterize the full automorphism groups of homogeneous and projective Reed-Muller (HRM and PRM) codes. These groups are respectively related to general and projective linear groups. Index Terms--Automorphism, projective linear group, projective Reed-Muller (PRM) codes.

A linear [[n, k, d].sub.q] code C is called near maximum-distance separable (NMDS) if d(C) = n - k and d([C.sup.[perpendicular to]]) = k. The maximum length of an NMDS [[n, k, d].sup.q] code is denoted by m'(k, q). In this correspondence, it has been verified by a computer-based proof that m'(5, 8) = 15, m'(4, 9) = 16, m'(5, 9) = 16, and 20 [less than or equal to] m'(4, 11) [less than or equal to] 21. Moreover, the NMDS codes of length m'(4.8), m'(5, 8), and m'(4, 9) have been classified. As the dual code of an NMDS code is NMDS, the values of m'(k, 8), k = 10, 11, 12, and of m'(k, 9),k = 12, 13, 14 have been also deduced. Index Terms--Galois fields, linear codes, near maximum-distance separable (NMDS) codes.

Let A(n, d) denote the maximum possible number of codewords in a binary code of length n and minimum Hamming distance d. For large values of n, the best known upper bound, for fixed d, is the Johnson bound. We give a new upper bound which is at least as good as the Johnson bound for all values of n and d, and for each d there are infinitely many values of n for which the new bound is better than the Johnson bound. For small values of n and d, the best known method to obtain upper bounds on A(n, d) is linear programming. We give new inequalities for the linear programming and show that with these new inequalities some of the known bounds on A(n, d) for n [less than or equal to] 28 are improved. Index Terms--A(n, d), holes, Johnson bound, linear programming bound.

Dougherty, Gaborit, Harada, Munemasa, and Sole have previously given an upper bound on the minimum Lee weight of a Type IV self-dual [Z.sub.4]-code, using a similar bound for the minimum distance of binary doubly even self-dual codes. We improve their bound, finding that the minimum Lee weight of a Type IV self-dual [Z.sub.4]-code of length n is at most 4[??]n/12[??], except when n = 4, and n = 8 when the bound is 4, and n = 16 when the bound is 8. We prove that the extremal binary doubly even self-dual codes of length n [greater than or equal to] 24, n [not equal to] 32 are not [Z.sub.4]-linear. We classify Type IV-I codes of length 16. We prove that all Type IV codes of length 24 have minimum Lee weight 4 and minimum Hamming weight 2, and the Euclidean-optimal Type IV-I codes of this length have minimum Euclidean weight 8. Index Terms--Type IV codes over rings, self-dual codes, [Z.sub.4]-linearity.

We prove that 23, 11, and 7 do not divide the order of the automorphism group of a binary [96, 48, 20] doubly-even self-dual code. We construct 25 new inequivalent binary [96, 48, 16] doubly-even self-dual codes via an automorphism of order 23. Index Terms--Automorphisms, doubly-even self-dual codes, weight enumerators.

In this correspondence, in extension of Piret's bound for codes over phase-shift keying (PSK) signal sets, we investigate the application of the Gilbert-Varshamov (GV) bound to a variety of distance-uniform (DU) signal sets in Euclidean space. It is shown that four-dimensional signal sets matched to binary tetrahedral, binary octahedral, and binary icosahedral groups lead to better bounds compared to the bounds for signal sets matched to dicyclic groups with the same number of signal points and comparable symmetric PSK signal sets. Index Terms--Euclidean space codes, Gilbert-Varshamov (GV) bound, group codes, multilevel construction, signal sets matched to groups, uniform signal sets.

Tailbiting codes encoded by feedback convolutional encoders are studied. A condition for when tailbiting will work is given and it is described how the encoder starting state can be obtained for feedback encoders in both controller and observer canonical forms. Finally, results from a search for systematic feedback encoders that encode tailbiting codes with good decoding bit error probabilities are presented. Index Terms--Convolutional codes, systematic feedback convolutional encoders, tailbiting trellis encoders.

We describe a simple technique for the numerical calculation, within any desired degree of accuracy, of the pairwise error probability (PEP) of space-time codes. This method applies also to the calculation of E[Q([square root of [xi]])] for any nonnegative random variable [xi] whose moment-generating function [[PHI].sub.[xi]](s) = E[exp(-s[xi])] is known. Index Terms--Multiple antennas, pairwise error probability (PEP), space-time codes.

Space-time block codes for providing transmit diversity in wireless communication systems are considered. Based on the principles of linearity and unitarity, a complete classification of linear codes is given in the case when the symbol constellations are complex, and the code is based on a square matrix or restriction of such by deleting columns (antennas). Maximal rate delay optimal codes are constructed within this category. The maximal rates allowed by linearity and unitarity fall off exponentially with the number of transmit antennas. Index Terms--Clifford algebras, multiple antennas, space-time block codes, transmit diversity.

We consider the problem of demodulating and decoding multiuser information symbols in an uplink asynchronous coded code-division multiple-access (CDMA) system employing long (aperiodic) spreading sequences, in the presence of unknown multipath channels, out-cell multiple-access interference (OMAI), and narrow-band interference (NBI). A blind turbo multiuser receiver, consisting of a novel blind Bayesian multiuser detector and a bank of MAP decoders, is developed for such a system. The effect of OMAI and NBI is modeled as colored Gaussian noise with some unknown covariance matrix. The main contribution of this paper is to develop blind Bayesian multiuser detectors for long-code multipath CDMA systems under both white and colored Gaussian noise. Such detectors are based on the Bayesian inference of all unknown quantities. The Gibbs sampler, a Markov chain Monte Carlo procedure, is then used to calculate the Bayesian estimates of the unknowns. The blind Bayesian multiuser detector computes the a posteriori probabilities of the channel coded symbols, which are differentially encoded before being sent to the channel. Being soft-input soft-output in nature, the proposed blind Bayesian multiuser detectors and the MAP decoders can iteratively exchange the extrinsic information to successively refine the performance, leading to the so-called blind turbo multiuser receiver. Index Terms--Coded system, Gibbs sampler, long-code CDMA, multipath, narrow-band interference, out-cell multiple-access interference, turbo multiuser detection.

We consider a space-time coded (STC) orthogonal frequency-division multiplexing (OFDM) system with multiple transmitter and receiver antennas over correlated frequency-and time-selective fading channels. It is shown that the product of the time-selectivity order and the frequency-selectivity order is a key parameter to characterize the outage capacity of the correlated fading channel. It is also observed that STCs with large effective lengths and ideal built-in interleavers are more effective in exploiting the natural diversity in multiple-antenna correlated fading channels. We then propose a low-density parity-check (LDPC)-code-based STC-OFDM system. Compared with the conventional space-time trellis code (STTC), the LDPC-based STC can significantly improve the system performance by exploiting both the spatial diversity and the selective-fading diversity in wireless channels. Compared with the recently proposed turbo-code-based STC scheme, LDPC-based STC exhibits lower receiver complexity and more flexible scalability. We also consider receiver design for LDPC-based STC-OFDM systems in unknown fast fading channels and propose a novel turbo receiver employing a maximum a posteriori expectation-maximization (MAP-EM) demodulator and a soft LDPC decoder, which can significantly reduce the error floor in fast fading channels with a modest computational complexity. With such a turbo receiver, the proposed LDPC-based STC-OFDM system is a promising solution to highly efficient data transmission over selective-fading mobile wireless channels. Index Terms--Correlated fading, iterative receiver, low-density parity-check (LDPC) codes, multiple antennas, orthogonal frequency-division multiplexing (OFDM), space-time code (STC).

A novel compound code is designed for the popular multidimensional (M-D) Wei trellis code [1] combined with a simple parity-check code. Using the iterative Viterbi decoding algorithm, we can achieve a remarkable performance improvement with low computational complexity. Simulation results show that at a bit error rate (BER) of 3.7 x [10.sup.-6] about 2.2-dB additional net gain has been obtained over the conventional scheme of the 4-D 16-state Wei code at a spectral efficiency of 6.7871 bit/T. Index Terms--Iterative Viterbi algorithm, multidimensional TCM, parity-concatenated trellis code.

We show that the state complexity profile of a convolutional code C is the same as that of the reciprocal of the dual code of C in case that minimal encoders for both codes are used. Then, we propose an optimum permutation for any given (n, n - 1) binary convolutional code that will yield an equivalent code with the lowest state complexity. With this permutation, we are able to find many (n, n - 1) binary convolutional codes which are better than punctured convolutional codes of the same code rate and memory size by either lower decoding complexity or better weight spectra. Index Terms--Convolutional codes, decoding, trellis codes.

Guided scrambling (GS) is used to control the runlength within codewords as well as to suppress dc components. A weakly constrained code designed by using a GS technique violates the imposed k-constraint with low probability, but achieves a code rate close to the channel capacity. In this study, the probability that codewords violate the k-constraint was measured by using convolutional GS and by using Galois-Field (GF)-addition GS. The occurrence probability of each runlength was also computed and then compared to a 24/25(0,8) block code having a high code rate and to the maxentropic runlength limited (RLL) sequence. The comparison shows that codes designed by these GS techniques attain extremely high code efficiency. Index Terms--Code efficiency, digital signal recording, guided scrambling, k-constraint, weakly constrained code.

In this correspondence, the statistical approach proposed by Agrawal and Vardy to evaluate the error performance of the generalized minimum distance (GMD) decoding is extended to other reliability-based decoding algorithms for binary linear block codes, namely Chase-type, combined GMD and Chase-type, and order statistic decodings (OSDs). In all cases, tighter and simpler bounds than those previously proposed have been obtained with this approach. Index Terms--Block codes, decoding, order statistics, reliability-based decoding, soft-decision decoding.

In an earlier work, Poor and Verdu established an upper bound for the reliability function of arbitrary single-user discrete-time channels with memory. They also conjectured that their bound is tight for all coding rates. In this note, we demonstrate via a counterexample involving memoryless binary erasure channels (BECs) that the Poor-Verdu upper bound is not tight at low rates. We conclude by examining possible improvements to this bound. Index Terms--Arbitrary channels with memory, binary erasure channels (BECs), channel coding, channel reliability function, information spectrum, probability of error.

In this correspondence, we present the results of the optimization applied to the design of interleavers for rate-1/n parallel concatenated convolutional codes (PCCC) tailored to specific recursive systematic convolutional (RSC) constituent codes. The emphasis is on low-latency codes associated with interleavers of block length less than or equal to 160. The error floors of the optimized codes are significantly lower than those associated with the use of random interleavers. The distance spectra of the equivalent block codes resulting from trellis termination applied to PCCC are evaluated and used to obtain asymptotic bit error rate (BER) curves for the optimized codes. Index Terms--Interleavers, iterative algorithms, optimization, permutations, turbo codes.

The synchronization recovery property of variable-length (VL) codes has been extensively studied. In the paper, the mean error propagation length (MEPL) and the variance of error propagation length (VEPL), which are the secondary performance criteria of a VL code, are introduced to measure the synchronization recovery capability of a VL code. For the same probability distribution, there exist many different VL codes which have the same redundancy as the Huffman code but quite different MEPLs and VEPLs. To find one of the VL codes which has the minimum MEPL is a very difficult problem. We present two design algorithms for finding minimum-redundancy VL codes with short MEPL and VEPL. These two algorithms are simple and have the property that the codewords are assigned one by one. The efficiency of the algorithms are tested extensively by comparing the algorithms with known construction methods available in literature. Actually, VL codes obtained by the two algorithms outperform almost all codes available. Index Terms--Error propagation length, synchronization, variable-length codes.

Tailbiting trellis representations of linear block codes with an arbitrary sectionalization of the time axis are studied. The notations of regular and irregular tailbiting codes are introduced and their maximal state complexities are lower-bounded. The asymptotic behavior of the derived bound is investigated. Furthermore, for regular tailbiting codes the product state complexity is lower-bounded. Tables of new tailbiting trellis representations of linear block codes of rates 1/2, 1/3, and 1/4 are presented. Almost all found trellises are optimal in the sense of the new bound on the state complexity and for most codes with nonoptimal trellises there exist time-varying trellises which are optimal. Five of our newly found tailbiting codes are better than the previously known linear codes with the same parameters. Four of them are also superior to any previously known nonlinear code with the same parameters. Also, more than 40 other quasi-cyclic codes have been found that improve the parameter set of previously known quasi-cyclic codes. Index Terms--Asymptotic complexity bounds, quasi-cyclic codes, state complexity, tailbiting codes, trellis.

The problem of determining the maximum size of a ternary code is considered, under the restriction that each symbol should appear a given number of times in each codeword. Upper and lower bounds on the size of such codes under Hamming metric are discussed, where the lower bounds follow from constructions of good codes. Some of the results are obtained by explicitly finding codes by computer search. A table of exact values and best known bounds on the maximum size for codes of length at most 10 is presented. Index Terms--Constant-weight codes, nonbinary codes, weight constraints.

We generalize constructions of lexicographic codes to produce locally optimal codes with a desired trellis decoding complexity. These constructions are efficient for high-rate codes and provide a means for automated code design. As a byproduct, we improve known bounds on the parameters of lexicodes. Index Terms--Code design, lexicographic codes (lexicodes), minimal trellises, trellis decoding.

Results of recent theoretical investigations highlighted that the use of chaos in direct-sequence code-division multiple access (DS-CDMA) systems may lead to nonnegligible improvements in communication quality for several scenarios. We here briefly review the main steps in this derivation and report the experimental verifications of the corresponding theoretical prediction. With this, we confirm that chaos-based spreading sequences outperform classical pseudo-random sequences in at least two important cases. Over nonselective channels, the ability of chaos-based spreading of minimizing multiple-access interference leads to a measured 60% improvement in [P.sub.err] with respect to classical spreading. Over selective channels, the possibility of jointly optimizing chaos-based spreading and rake receiver profiles leads to improvements of up to 38% in [P.sub.err] with respect to systems with either conventional spreading or conventional rake policies. Index Terms--Chaos, code-division multiple access, communication-system performance, rake receiver.

We address the problem of designing jointly optimum linear precoder and decoder for a MIMO channel possibly with delay-spread, using a weighted minimum mean-squared error (MMSE) criterion subject to a transmit power constraint. We show that the optimum linear precoder and decoder diagonalize the MIMO channel into eigen subchannels, for any set of error weights. Furthermore, we derive the optimum linear precoder and decoder as functions of the error weights and consider specialized designs based on specific choices of error weights. We show how to obtain: 1) the maximum information rate design; 2) QoS-based design (we show how to achieve any set of relative SNRs across the subchannels); and 3) the (unweighted) MMSE and equal-error design for fixed rate systems. Index Terms--MIMO, MMSE, precoding, space-time coding, spatial multiplexing.

Orthogonal frequency-division multiplexing (OFDM) is a popular line code scheme that has found application in wideband modern technologies such has asynchronous digital subscriber loop, high bit rate digital subscriber loop, in addition to digital audio broadcast, digital television terrestrial broadcast, and mobile data communication. To prevent interblock interference (IBI), OFDM inserts a guard time sequence, called the cyclic prefix (CP), between each transmitted block. In practice, CP is rarely long enough for the complete elimination of IBI. Understanding the nature of IBI is crucial in subduing its adverse effects. This study characterizes the power spectral density of the IBI in terms of the channel impulse response and the cyclic prefix length. One of the key findings is that the samples of the impulse response tails do not contribute to IBI power uniformly, but rather their contributions grow linearly as the samples move closer to the edges of impulse response. Based on this observation, the study proposes a time of reference computation technique that minimizes the total IBI power as seen by the receiver. Index Terms--Interblock interference, orthogonal frequency-division multiplexing (OFDM), time-domain equalizer.

Improved space-time coding for multiple-input and multiple-output orthogonal frequency division multiplexing is studied for wireless systems using QPSK modulation for four transmit and four receive antennas. A 256-state code is shown to perform within 3 dB of outage capacity (and within 2 dB with perfect channel estimation), which is better than any other published result without using iterative decoding. Index Terms--Antenna diversity, MIMO, OFDM, space-time codes.

Serially concatenated and interleaved continuous phase modulation (CPM) with iterative decoding is investigated. An a posteriori probability (APP) algorithm for CPM is developed based on the classic APP algorithm for channel codes. The system is analyzed through upper bounds on the average bit error probability. For coded and interleaved minimum shift keying, the weight spectrum is computed, resulting in a transfer function bound. This is cumbersome for a general CPM system; instead, only the most significant error events contributing to the weight spectrum are identified. Simulations show that, firstly, these events give a satisfactory view of system performance when equal outer codes are used, and secondly, that remarkably good performance can be obtained for some simple systems. Finally, power spectral densities and bandwidths are computed, allowing for a bandwidth/performance comparison of different combinations. Index Terms--APP algorithm, concatenated codes, continuous phase modulation, iterative decoding, serial concatenation.

This paper develops an approach to iterative multistage decoding of multilevel codes. This involves passing reliability information to previous and subsequent decoders instead of only hard decisions to subsequent decoders. The paper also develops an adaptive version of the suboptimal soft output decoding algorithm of [1]. This adaptive algorithm provides a gain of approximately 0.24 dB at a bit error rate (BER) of [10.sup.-5] after four iterations and approximately 0.43 dB after ten iterations over the algorithm of [1]. If the adaptive algorithm is used in conjunction with iterative multistage decoding then a gain of approximately 0.62 dB is obtained at a BER of [10.sup.-5] after four iterations and approximately 0.9 dB after ten iterations over the algorithm of [1]. Index Terms--Block turbo codes, iterative decoding, iterative multistage decoding, multilevel codes, product codes.

This letter studies the numerical properties of Log-MAP-based soft-input soft-output decoding. The ranges of the differences between the path metrics, the soft output, and the complete information are derived. Since the Log-MAP algorithm is functionally dependent only upon the differences between path metrics, not their magnitudes, a modulo normalization technique that is automatically implemented in two's complement arithmetic can be used to accommodate overflow of the path metrics. An expression for the minimum internal data width of a soft-input soft-output decoder that employs the modulo normalization technique is derived. As an application, the minimum internal data width of a decoder for the turbo code used in third-generation mobile communication systems is determined. Index Terms--Iterative decoding, maximum a posteriori algorithm, soft-input soft-output decoder, turbo codes.