Showing papers on "List decoding published in 1996"

Expander codes

Abstract: Using expander graphs, we construct a new family of asymptotically good, linear error-correcting codes. These codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors. We present both randomized and explicit constructions of these codes. Experimental results demonstrate the good performance of the randomly chosen codes.

Soft-Output Decoding Algorithms in Iterative Decoding of Turbo Codes

Abstract: In this article, we present two versions of a simplified maximum a posteriori decoding algorithm. The algorithms work in a sliding window form, like the Viterbi algorithm, and can thus be used to decode continuously transmitted sequences obtained by parallel concatenated codes, without requiring code trellis termination. A heuristic explanation is also given of how to embed the maximum a posteriori algorithms into the iterative decoding of parallel concatenated codes (turbo codes). The performances of the two algorithms are compared on the basis of a powerful rate 1/3 parallel concatenated code. Basic circuits to implement the simplified a posteriori decoding algorithm using lookup tables, and two further approximations (linear and threshold), with a very small penalty, to eliminate the need for lookup tables are proposed.

A linear time erasure-resilient code with nearly optimal recovery

Abstract: We develop an efficient scheme that produces an encoding of a given message such that the message can be decoded from any portion of the encoding that is approximately equal to the length of the message. More precisely, an (n,c,l,r)-erasure-resilient code consists of an encoding algorithm and a decoding algorithm with the following properties. The encoding algorithm produces a set of l-bit packets of total length cn from an n-bit message. The decoding algorithm is able to recover the message from any set of packets whose total length is r, i.e., from any set of r/l packets. We describe erasure-resilient codes where both the encoding and decoding algorithms run in linear time and where r is only slightly larger than n.

Iterative decoding of serially concatenated convolutional codes

Abstract: Serial concatenation of convolutional codes separated by an interleaver has recently been shown, through the use of upper bounds to the maximum likelihood performance, to be competitive with parallel concatenated coding schemes known in the literature as ‘turbo codes’. The most important feature of turbo codes consists in their relatively simple, yet high performance, iterative decoding algorithm. The authors propose a new iterative decoding algorithm for serial concatenation, and show that the new coding scheme can yield a significant advantage with respect to turbo codes.

Fast generalized minimum-distance decoding of algebraic-geometry and Reed-Solomon codes

Abstract: Generalized minimum-distance (GMD) decoding is a standard soft-decoding method for block codes. We derive an efficient general GMD decoding scheme for linear block codes in the framework of error-correcting pairs. Special attention is paid to Reed-Solomon (RS) codes and one-point algebraic-geometry (AG) codes. For RS codes of length n and minimum Hamming distance d the GMD decoding complexity turns out to be in the order O(nd), where the complexity is counted as the number of multiplications in the field of concern. For AG codes the GMD decoding complexity is highly dependent on the curve in consideration. It is shown that we can find all relevant error-erasure-locating functions with complexity O(o/sub 1/nd), where o/sub 1/ is the size of the first nongap in the function space associated with the code. A full GMD decoding procedure for a one-point AG code can be performed with complexity O(dn/sup 2/).

Soft-output decoding algorithms for continuous decoding of parallel concatenated convolutional codes

Abstract: We propose new decoding algorithms to be embedded in the iterative decoding strategy of parallel concatenated convolutional codes. They are derived from the optimum maximum-a-posteriori algorithm and permit a continuous decoding of the coded sequence without requiring trellis termination of the constituent codes. Two basic versions of the continuous algorithm and their suboptimum simplifications are described. Simulation results refer to the applications of the new algorithms to a highly efficient rate 1/3 concatenated code; they show performance only 0.6 dB worse than the Shannon limit.

Introduction to Coding and Information Theory

Abstract: This book is intended to introduce coding theory and information theory to undergraduate students of mathematics and computer science. It begins with a review of probablity theory as applied to finite sample spaces and a general introduction to the nature and types of codes. The two subsequent chapters discuss information theory: efficiency of codes, the entropy of information sources, and Shannon's Noiseless Coding Theorem. The remaining three chapters deal with coding theory: communication channels, decoding in the presence of errors, the general theory of linear codes, and such specific codes as Hamming codes, the simplex codes, and many others.

The Cryptographic Security of the Syndrome Decoding Problem for Rank Distance Codes

Abstract: We present an algorithm that achieves general syndrome decoding of a (n, k, r) linear rank distance code over GF(q m ) in O(nr + m)3q(m−r)(r−1)) elementary operations. As a consequence, the cryptographical schemes [Che94, Che96] which rely on this problem are not secure with the proposed parameters. We also derive from our algorithm a bound on the minimal rank distance of a linear code which shows that the parameters from [Che94] are inconsistent.

Generalized minimum-distance decoding of Euclidean-space codes and lattices

Abstract: It is shown that multistage generalized minimum-distance (GMD) decoding of Euclidean-space codes and lattices can provide an excellent tradeoff between performance and complexity. We introduce a reliability metric for Gaussian channels that is easily computed from an inner product, and prove that a multistage GMD decoder using this metric is a bounded-distance decoder up to the true packing radius. The effective error coefficient of multistage GMD decoding is determined. Two simple modifications in the GMD decoding algorithm that drastically reduce this error coefficient are proposed. It is shown that with these modifications GMD decoding achieves the error coefficient of maximum-likelihood decoding for block codes and for generalized construction A lattices. Multistage GMD decoding of the lattices D/sub 4/, E/sub 8/, K/sub 12/, BW/sub 16/, and /spl Lambda//sub 24/ is investigated in detail. For K/sub 12/BW/sub 16/, and /spl Lambda//sub 24/, the GMD decoders have considerably lower complexity than the best known maximum-likelihood or bounded-distance decoding algorithms, and appear to be the most practically attractive decoders available. For high-dimensional codes and lattices (/spl ges/64 dimensions) maximum-likelihood decoding becomes infeasible, while GMD decoding algorithms remain quite practical. As an example, we devise a multistage GMD decoder for a 128-dimensional sphere packing with a nominal coding gain of 8.98 dB that attains an effective error coefficient of 1365760. This decoder requires only about 400 real operations, in addition to algebraic errors-and-erasures decoding of certain BCH and Hamming codes. It therefore appears to be practically feasible to implement algebraic multistage GMD decoders for high-dimensional sphere packings, and thus achieve high effective coding gains.

Performance of Reed-Solomon block turbo code

Abstract: Thanks to recent progress in the iterative decoding of concatenated codes, several new fields of investigation have appeared. In this paper, we present a first approach of the iterative decoding of Reed-Solomon (RS) product codes: "Turbo codes RS". Two methods to construct RS product codes are given. The iterative decoding of the RS product codes is based on the soft decoding and the soft decision of the component codes. The performance of RS turbo codes have been evaluated on the Gaussian and Rayleigh channels using Monte Carlo simulation. Coding gains up to 5.5 dB for a BER (bit error rate) of 10/sup -5/ have been obtained on the Gaussian channel. This new coding scheme is very attractive for data storage applications where the RS product codes are often used.

Computationally efficient soft-decision decoding of linear block codes based on ordered statistics

Abstract: Soft-decision decoding of a linear block code using the most reliable basis corresponding to each received word is investigated. Based either on probabilistic properties or on the structure of the code considered, three improvements to the algorithm devised by Fossorier and Lin (see ibid., vol.41, no.9, p.1379-1396, 1995) are presented. These modifications allow large computation savings or significant decoding speedup with little error performance degradation. First, a reduced probabilistic list of codeword candidates is associated with order-i reprocessing of a given code. It results in a large reduction of the maximum number of computations with a very small degradation in performance. Then, a probabilistic stopping criterion is introduced for order-0 reprocessing. This new test significantly decreases the average number of computations when appropriately implemented. Finally, the application of the algorithm to coset decoding is considered for |u|u+v| constructed codes. In addition to the conventional coset decoding, a new adaptive practically optimum coset decoding method is presented where at each reprocessing stage, the number of surviving cosets decreases. Suboptimum closest coset decoding is also investigated. It is shown that two-stage decoding with the algorithm of Fossorier and Lin offers a large variety of choices, since the reprocessing order of each stage can be determined independently.

Algorithm for continuous decoding of turbo codes

Abstract: A new continuous version of the maximum a posteriori algorithm is described and applied to sequence oriented decoding of parallel concatenated convolutional codes

Trellis decoding complexity of linear block codes

Abstract: 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.

A* decoding of block codes

Abstract: The A* algorithm is applied to maximum-likelihood soft-decision decoding of binary linear block codes. This paper gives a tutorial on the A* algorithm, compares the decoding complexity with that of exhaustive search and Viterbi decoding algorithms, and presents performance curves obtained for several codes.

On the implementation of minimum-redundancy prefix codes

Abstract: Minimum-redundancy coding (also known as Huffman (1952) coding) is one of the enduring techniques of data compression. We examine how best minimum-redundancy coding can be implemented, with particular emphasis on the situation when n is large, perhaps of the order of 10/sup 6/. We review techniques for devising minimum-redundancy codes, and consider in detail how encoding and decoding should be accomplished. In particular, we describe a modified decoding method that allows improved decoding throughput, requiring just a few machine operations per output symbol (rather than for each decoded bit), and uses just a few hundred bytes of memory above and beyond the space required to store an enumeration of the source alphabet. We review methods for calculating codeword lengths, show how those codeword lengths should be used to derive a minimum-redundancy code that has the alphabetic sequence property, and describes a memory-compact method for decoding such canonical codes. An improved method for decoding canonical codes is also presented.

The shift bound for cyclic, Reed-Muller and geometric Goppa codes

Abstract: We give a generalization of the shift bound on the minimum distance for cyclic codes which applies to Reed-Muller and algebraicgeometric codes. The number of errors one can correct by majority coset decoding is up to half the shift bound.

Efficient bounded-distance decoding of the hexacode and associated decoders for the Leech lattice and the Golay code

Abstract: Two soft-decision decoding algorithms for the (6, 3, 4) quaternary code hexacode are presented. Both algorithms realize half the minimum Euclidean distance of the code. The proposed algorithms are most practical. In using them, bounded-distance decoding of the Golay code and the Leech lattice are performed with at most 187 and 519 real-number operations respectively. Compare this to 651, respectively 3595, operations required by the best known maximum likelihood decoders (Vardy and Be'ery, 1991, 1993), and 431, respectively 1007, operations required by the bounded-distance decoders (Amrani et al., 1994). We present some simulation results for the proposed Leech lattice decoders revealing near-optimal performance. A comparison to known trellis codes is also provided.

Minimal trellis design for linear codes based on the Shannon product

Abstract: A novel trellis design technique for both block and convolutional codes based on the Shannon (1956) product of component block codes is introduced. Using the proposed technique, structured trellises for block and convolutional codes have been designed. It is shown that the designed trellises are minimal and allow reduced complexity Viterbi decoding.

Suboptimal decoding of linear codes: partition technique

Abstract: General symmetric channels are introduced, and near-maximum-likelihood decoding in these channels is studied. First, we define a class of suboptimal decoding algorithms based on an incomplete search through the code trellis. It is proved that the decoding error probability of suboptimal decoding is bounded above for any q-ary code of length n and code rate r by twice the error probability of its maximum-likelihood decoding and tends to the latter as n grows. Second, we design a suboptimal trellis-like algorithm, which reduces the known decoding complexity of the order of q/sup n min (r,1-r)/ operations to that of q/sup nr(i-r)/ operations for all cyclic codes and virtually all long linear codes. We also consider the corresponding bounds for concatenated codes. An important corollary is that this suboptimal decoding can provide complexity below the lower bounds on trellis complexity at a negligible expense in terms of decoding error probability.

Group codes generated by finite reflection groups

Abstract: Slepian-type group codes generated by finite Coxeter groups are considered. The resulting class of group codes is a generalization of the well-known permutation modulation codes of Slepian (1965), it is shown that a restricted initial-point problem for these codes has a canonical solution that can easily be computed. This allows one to enumerate all optimal group codes in this restricted sense and essentially solves the initial-point problem for all finite reflection groups. Formulas for the cardinality and the minimum distance of such codes are given. The new optimal group codes from exceptional reflection groups that are obtained achieve high rates and have excellent distance properties. The decoding regions for maximum-likelihood (ML) decoding are explicitly characterized and an efficient ML-decoding algorithm is presented. This algorithm relies on an extension of Slepian's decoding of permutation modulation and has similar low complexity,.

Sequential decoding using a priori information

Abstract: A metric for sequential decoding, based on the well known Fano metric, is proposed. It is suitable for using a priori information about the source bit probability in addition to soft inputs. The advantage of this approach is a considerable reduction in the achievable bit error rate (BER) and the corresponding computational complexity of the sequential decoding algorithm (Pareto distribution). Furthermore, channel state information can easily be taken into account in this metric by applying log-likelihood ratios. Additional improvements are possible for systematic convolutional codes. Simulation results are presented for two different binary sources.

Coding and decoding apparatus which transmits and receives tool information for constructing decoding scheme

Abstract: A coding and decoding apparatus is constructed so that the coding side transmits coded data together with identifying information for identifying the device of decoding the coded data, and the decoding side is capable of storing a number of decoding schemes so as to perform decoding based on one of the previously stored schemes. The apparatus further has devices for storing the received tools and tool-correspondent information which numerically represents the capacities of the tools so that it can make a comparison between the decoding capacity and the processing capacities of the tools to determine the possibility of the operations of the received tools. Further, a set of the tools are hierarchized so that the coded data produced by the n-ranked tool can be decoded by the (n+1)-ranked tool. Alternatively, the tools are defined in a hierarchical manner so that the decoding tools installed in the decoding apparatus will be able to assure the minimum quality and the requested decoding process can be performed by the received decoding tool. Further, the identification code of the decoding scheme used can be transmitted as required so that the decoding scheme can be expanded by transmitting the differential information from the basic decoding scheme.

On classes of rate k/(k+1) convolutional codes and their decoding techniques

Abstract: For the class of rate k/(k+1) convolutional codes, Yamada et al. (1983) proposed an efficient maximum-likelihood decoding algorithm called the YHM algorithm. In order to reduce the complexity of the YHM algorithm, this paper presents two techniques for simplifying the trellis diagram used in the YHM algorithm. We further observe that the proposed techniques effectively reduce the complexity of the YHM algorithm for two classes /spl Xi/ and /spl Xi//sub f/ (which is a subclass of /spl Xi/) of rate k/(k+1) convolutional codes. The construction of codes in these classes is also discussed. It is shown that /spl Xi/ codes with d/sub free/=3,4 can be obtained by simple construction. A code search algorithm for /spl Xi/ codes with d/sub free//spl ges/5 is also introduced. Computer searches are performed to construct good /spl Xi/ and /spl Xi//sub f/ codes. For specified decoding complexities, a number of these new codes give better error performance than previously reported codes.

Fast decoding of tagged message formats

Abstract: Many important protocols, such as Q.2931 or any protocol based on the ASN.1 basic encoding rules, are transmitted using tagged message formats, in which a message can be considered as a sequence of interleaved tag and data fields, where tag fields define the meaning of subsequent fields. These messages are computationally expensive to decode, partly because decoding each data field requires resting one or more tag fields. Evidence suggests that in some applications, although the potential space of message encodings may be very large, only a small number of message layouts are seen frequently, and thus some of the work required in decoding can be amortized over many messages. This paper analyzes the use of run-time code generation to generate optimized decoding instruction sequences for received messages matching previously observed layouts, and describes a prototype system that applies the techniques to decoding the Q.2931 and ASN.1 BER protocols. In the average case, substantial performance gains are seen.

Variable-complexity trellis decoding of binary convolutional codes

Abstract: Considers trellis decoding of convolutional codes with selectable effort, as measured by decoder complexity. Decoding is described for single parent codes with a variety of complexities, with performance "near" that of the optimal fixed receiver complexity coding system. Effective free distance is examined. Criteria are proposed for ranking parent codes, and some codes found to be best according to the criteria are tabulated, Several codes with effective free distance better than the best code of comparable complexity were found. Asymptotic (high SNR) performance analysis and error propagation are discussed. Simulation results are also provided.

Encoding of dklr-sequences using one weight set

Abstract: Traditional schemes for encoding and decoding runlength-constrained sequences using the enumeration principle require two sets of weighting coefficients. A new enumeration is presented requiring only one set of coefficients.

New decoder for double-error-correcting binary BCH codes

Abstract: A step-by-step decoding algorithm is presented for double-error-correcting binary BCH codes of length n=2/sup m/-1. The decoding algorithm can directly determine whether any received bit is correct or not without knowing the number of errors which have occurred in the received vector and also without determining the corresponding error location polynomial. The merits of the hardware decoder based on the algorithm are that it is very simple and fast. In addition, it is suitable for decoding long block codes.

Majority-logic-like decoding of vector symbols

Abstract: The use of the structure of one-step decodable majority logic codes for enhanced and simplified vector symbol decoding, such as outer code decoding of concatenated codes, is proposed. For J equations checking a particular symbol, the technique to be described almost always corrects the symbol if there are J-1 or fewer symbol errors, and often corrects cases of far more than J symbol errors. Ordinarily, majority level decoding with J equations for a symbol corrects the symbol in all cases where there are up to [J/2] errors. The decoding power is comparable to Reed-Solomon codes, but decoding is simpler than for Reed-Solomon codes.

Fast Decoding Algorithms for First Order Reed-Muller and Related Codes

Abstract: Fast decoding algorithms for short codes based on modifications of maximum likelihood decoding algorithms of first order Reed-Muller codes are described. Only additions-subtractions, comparisons and absolute value calculations are used in the algorithms. Soft and hard decisions maximum likelihood decoding algorithms for first order Reed-Muller and the Nordstrom-Robinson codes with low complexity are proposed.

Note on decoding binary goppa codes

Abstract: The author gives a simple expression for the polynomial y(x) which solves the polynomial equation y(x)/sup 2//spl equiv/t(x) mod G(x), where t(x), y(x) and G(x) are polynomials over the field GF(2/sup m/). The solution of such an equation is a step in the so called Patterson algorithm for decoding binary Goppa codes. The result may also be useful for other applications.