Linear-time encodable/decodable codes with near-optimal rate

We present an explicit construction of linear-time encodable and decodable codes of rate r which can correct a fraction (1-r-/spl epsiv/)/2 of errors over an alphabet of constant size depending only on /spl epsiv/, for every 00. The error-correction performance of these codes is optimal as seen by the Singleton bound (these are "near-MDS" codes). Such near-MDS linear-time codes were known for the decoding from erasures; our construction generalizes this to handle errors as well. Concatenating these codes with good, constant-sized binary codes gives a construction of linear-time binary codes which meet the Zyablov bound, and also the more general Blokh-Zyablov bound (by resorting to multilevel concatenation). Our work also yields linear-time encodable/decodable codes which match Forney's error exponent for concatenated codes for communication over the binary symmetric channel. The encoding/decoding complexity was quadratic in Forney's result, and Forney's bound has remained the best constructive error exponent for almost 40 years now. In summary, our results match the performance of the previously known explicit constructions of codes that had polynomial time encoding and decoding, but in addition have linear-time encoding and decoding algorithms.     

Soft-decision decoding of Reed-Muller codes: a simplified algorithm

Soft-decision decoding is considered for general Reed-Muller (RM) codes of length n and distance d used over a memoryless channel. A recursive decoding algorithm is designed and its decoding threshold is derived for long RM codes. The algorithm has complexity of order nlnn and corrects most error patterns of the Euclidean weight of order radicn/lnn, instead of the decoding threshold radicd/2 of the bounded distance decoding. Also, for long RM codes of fixed rate R, the new algorithm increases 4/pi times the decoding threshold of its hard-decision counterpart     

Asymptotic properties on codeword lengths of an optimal FV code for general sources

This correspondence is concerned with asymptotic properties on the codeword length of a fixed-to-variable length code (FV code) for a general source {X/sup n/}/sub n=1//sup /spl infin// with a finite or countably infinite alphabet. Suppose that for each n /spl ges/ 1 X/sup n/ is encoded to a binary codeword /spl phi//sub n/(X/sup n/) of length l(/spl phi//sub n/(X/sup n/)). Letting /spl epsiv//sub n/ denote the decoding error probability, we consider the following two criteria on FV codes: i) /spl epsiv//sub n/ = 0 for all n /spl ges/ 1 and ii) lim sup/sub n/spl rarr//spl infin///spl epsiv//sub n/ /spl les/ /spl epsiv/ for an arbitrarily given /spl epsiv/ /spl isin/ [0,1). Under criterion i), we show that, if X/sup n/ is encoded by an arbitrary prefix-free FV code asymptotically achieving the entropy, 1/nl(/spl phi//sub n/(X/sup n/)) - 1/nlog/sub 2/ 1/PX/sup n/(X/sup n/) /spl rarr/ 0 in probability as n /spl rarr/ /spl infin/ under a certain condition, where P/sub X//sup n/ denotes the probability distribution of X/sup n/. Under criterion ii), we first determine the minimum rate achieved by FV codes. Next, we show that 1/nl(/spl phi//sub n/(X/sup n/)) of an arbitrary FV code achieving the minimum rate in a certain sense has a property similar to the lossless case.     

Decoding of Expander Codes at Rates Close to Capacity

The decoding error probability of codes is studied as a function of their block length. It is shown that the existence of codes with a polynomially small decoding error probability implies the existence of codes with an exponentially small decoding error probability. Specifically, it is assumed that there exists a family of codes of length N and rate R=(1-epsiv)C (C is a capacity of a binary-symmetric channel), whose decoding probability decreases inverse polynomially in N. It is shown that if the decoding probability decreases sufficiently fast, but still only inverse polynomially fast in N, then there exists another such family of codes whose decoding error probability decreases exponentially fast in N. Moreover, if the decoding time complexity of the assumed family of codes is polynomial in N and 1/epsiv, then the decoding time complexity of the presented family is linear in N and polynomial in 1/epsiv. These codes are compared to the recently presented codes of Barg and Zemor, "Error Exponents of Expander Codes", IEEE Transactions on Information Theory, 2002, and "Concatenated Codes: Serial and Parallel", IEEE Transactions on Information Theory, 2005. It is shown that the latter families cannot be tuned to have exponentially decaying (in N) error probability, and at the same time to have decoding time complexity linear in N and polynomial in 1/epsiv     

List decoding from erasures: bounds and code constructions

We consider the problem of list decoding from erasures. We establish lower and upper bounds on the rate of a (binary linear) code that can be list decoded with list size L when up to a fraction p of its symbols are adversarially erased. Such bounds already exist in the literature, albeit under the label of generalized Hamming weights, and we make their connection to list decoding from erasures explicit. Our bounds show that in the limit of large L, the rate of such a code approaches the "capacity" (1 - p) of the erasure channel. Such nicely list decodable codes are then used as inner codes in a suitable concatenation scheme to give a uniformly constructive family of asymptotically good binary linear codes of rate /spl Omega/(/spl epsiv//sup 2//log(1//spl epsiv/)) that can be efficiently list-decoded using lists of size O(1//spl epsiv/) when an adversarially chosen (1 - /spl epsiv/) fraction of symbols are erased, for arbitrary /spl epsiv/ > 0. This improves previous results in this vein, which achieved a rate of /spl Omega/(/spl epsiv//sup 3/log(1//spl epsiv/)).     

List decoding of q-ary Reed-Muller codes

The q-ary Reed-Muller (RM) codes RM/sub q/(u,m) of length n=q/sup m/ are a generalization of Reed-Solomon (RS) codes, which use polynomials in m variables to encode messages through functional encoding. Using an idea of reducing the multivariate case to the univariate case, randomized list-decoding algorithms for RM codes were given in and . The algorithm in Sudan et al. (1999) is an improvement of the algorithm in , it is applicable to codes RM/sub q/(u,m) with u相似文献   

Computing the error linear complexity spectrum of a binary sequence of period 2/sup n/

Binary sequences with high linear complexity are of interest in cryptography. The linear complexity should remain high even when a small number of changes are made to the sequence. The error linear complexity spectrum of a sequence reveals how the linear complexity of the sequence varies as an increasing number of the bits of the sequence are changed. We present an algorithm which computes the error linear complexity for binary sequences of period /spl lscr/=2/sup n/ using O(/spl lscr/(log/spl lscr/)/sup 2/) bit operations. The algorithm generalizes both the Games-Chan (1983) and Stamp-Martin (1993) algorithms, which compute the linear complexity and the k-error linear complexity of a binary sequence of period /spl lscr/=2/sup n/, respectively. We also discuss an application of an extension of our algorithm to decoding a class of linear subcodes of Reed-Muller codes.     

Throughput analysis for code division multiple accessing of the spread spectrum channel

We consider the use of error correction codes of rate r on top of pseudonoise (PN) sequence coding for code division multiple accessing of the spread spectrum channel. The channel is found to have a maximum throughput of 0.72 and 0.36 based on the evaluation of channel capacity and cutoff rate, respectively. More generally, these two values are derived for given bandwidth expanding n/r versus n/N where n is the length of the PN sequence and N is the number of Simultaneous users. It is found that to achieve the maximum throughput, n should be small. This implies that coding schemes with short PN sequences and low rate codes are superior in terms of throughput or antijam capability. The extreme case of n = 1 corresponds to using a very low rate code with no PN sequence coding. Convolutional codes are recommended and analyzed for their error rate and decoding complexity.     

Digital fingerprinting codes: problem statements, constructions, identification of traitors

We consider a general fingerprinting problem of digital data under which coalitions of users can alter or erase some bits in their copies in order to create an illegal copy. Each user is assigned a fingerprint which is a word in a fingerprinting code of size M (the total number of users) and length n. We present binary fingerprinting codes secure against size-t coalitions which enable the distributor (decoder) to recover at least one of the users from the coalition with probability of error exp(-/spl Omega/(n)) for M=exp(/spl Omega/(n)). This is an improvement over the best known schemes that provide the error probability no better than exp(-/spl Omega/(n/sup 1/2/)) and for this probability support at most exp(O(n/sup 1/2/)) users. The construction complexity of codes is polynomial in n. We also present versions of these constructions that afford identification algorithms of complexity poly(n)=polylog(M), improving over the best previously known complexity of /spl Omega/(M). For the case t=2, we construct codes of exponential size with even stronger performance, namely, for which the distributor can either recover both users from the coalition with probability 1-exp(/spl Omega/(n)), or identify one traitor with probability 1.     

Stopping set distribution of LDPC code ensembles

Stopping sets determine the performance of low-density parity-check (LDPC) codes under iterative decoding over erasure channels. We derive several results on the asymptotic behavior of stopping sets in Tanner-graph ensembles, including the following. An expression for the normalized average stopping set distribution, yielding, in particular, a critical fraction of the block length above which codes have exponentially many stopping sets of that size. A relation between the degree distribution and the likely size of the smallest nonempty stopping set, showing that for a /spl radic/1-/spl lambda/'(0)/spl rho/'(1) fraction of codes with /spl lambda/'(0)/spl rho/'(1)<1, and in particular for almost all codes with smallest variable degree >2, the smallest nonempty stopping set is linear in the block length. Bounds on the average block error probability as a function of the erasure probability /spl epsi/, showing in particular that for codes with lowest variable degree 2, if /spl epsi/ is below a certain threshold, the asymptotic average block error probability is 1-/spl radic/1-/spl lambda/'(0)/spl rho/'(1)/spl epsi/.     

Error Exponents for Recursive Decoding of Reed&#8211;Muller Codes on a Binary-Symmetric Channel

Error exponents are studied for recursive decoding of Reed-Muller (RM) codes and their subcodes used on a binary-symmetric channel. The decoding process is first decomposed into similar steps, with one new information bit derived in each step. Multiple recursive additions and multiplications of the randomly corrupted channel outputs plusmn1 are performed using a specific order of these two operations in each step. Recalculated random outputs are compared in terms of their exponential moments. As a result, tight analytical bounds are obtained for decoding error probability of the two recursive algorithms considered in the paper. For both algorithms, the derived error exponents almost coincide with simulation results. Comparison of these bounds with similar bounds for bounded distance decoding and majority decoding shows that recursive decoding can reduce the output error probability of the latter two algorithms by five or more orders of magnitude even on the short block length of 256. It is also proven that the error probability of recursive decoding can be exponentially reduced by eliminating one or a few information bits from the original RM code     

一种基于奇偶校验码级联极化码的低复杂度译码算法

极化码作为一种纠错码,具有较好的编译码性能,已成为5G短码控制信道的标准编码方案.但在码长较短时,其性能不够优异.作为一种新型级联极化码,奇偶校验码与极化码的级联方案提高了有限码长的性能,但是其译码算法有着较高的复杂度.该文针对这一问题,提出一种基于奇偶校验码级联极化码的串行抵消局部列表译码(PC-PSCL)算法,该算...     

Error Exponents for Two Soft-Decision Decoding Algorithms of Reed–Muller Codes

Error exponents are studied for recursive and majority decoding of general Reed–Muller (RM) codes ${rm RM}(r,m)$ used on the additive white Gaussian noise (AWGN) channels. Both algorithms have low complexity and correct many error patterns whose weight exceeds half the code distance. Decoding consists of multiple consecutive steps, which repeatedly recalculate the input symbols and determine different information symbols using soft-decision majority voting. For any code ${rm RM}(r,m)$, we estimate the probabilities of the information symbols obtained in these recalculations and derive the analytical upper bounds for the block error rates of the recursive and majority decoding. In the case of a low noise, we also obtain the lower bounds and show that the upper bounds are tight. For a higher noise, these bounds closely approach our simulation results.      

Soft-decision decoding using punctured codes

Let a q-ary linear (n,k)-code be used over a memoryless channel. We design a soft-decision decoding algorithm that tries to locate a few most probable error patterns on a shorter length s ∈ [k,n]. First, we take s cyclically consecutive positions starting from any initial point. Then we cut the subinterval of length s into two parts and examine T most plausible error patterns on either part. To obtain codewords of a punctured (s,k)-code, we try to match the syndromes of both parts. Finally, the designed codewords of an (s,k)-code are re-encoded to find the most probable codeword on the full length n. For any long linear code, the decoding error probability of this algorithm can be made arbitrarily close to the probability of its maximum-likelihood (ML) decoding given sufficiently large T. By optimizing s, we prove that this near-ML decoding can be achieved by using only T≈q^(n-k)k(n+k)/ error patterns. For most long linear codes, this optimization also gives about T re-encoded codewords. As a result, we obtain the lowest complexity order of q^(n-k)k(n+k)/ known to date for near-ML decoding. For codes of rate 1/2, the new bound grows as a cubic root of the general trellis complexity q^min{n-k,k}. For short blocks of length 63, the algorithm reduces the complexity of the trellis design by a few decimal orders     

Search and determination of convolutional self-doubly orthogonal codes for iterative threshold decoding

In this paper, we present new results on the search and determination of wide-sense convolutional self-doubly orthogonal codes (CSO/sup 2/C-WS) which can be decoded using a simple iterative threshold decoding algorithm without interleaving. For their iterative decoding, in order to ensure the independence of observables over the first two iterations without the presence of interleavers, these CSO/sup 2/C must satisfy specific orthogonal properties of their generator connections. The error performances of CSO/sup 2/C, depend essentially on the number of taps J of the code generators but not on the code memory length. Since the overall latency of the iterative threshold decoding process is proportional to the memory length of the codes, therefore, when searching for the best CSO/sup 2/C-WS of a given J value, the memory length of the codes should be chosen to be as small as possible. In this paper, we present a code-searching technique based on heuristic computer searching algorithms which have yielded the best known CSO/sup 2/C-WS. The construction method for CSO/sup 2/C-WS has provided the best known r=1/2 codes with the shortest memory length having J/spl les/30. Although not very complex to implement, the search method presented here is quite efficient especially in reducing very substantially the execution time required to determine the codes with the shortest spans. Furthermore, in addition to presenting the search results for the codes, error performances obtained by simulation are also provided.     

Box and match techniques applied to soft-decision decoding

In this paper, we improve the ordered statistics decoding algorithm by using matching techniques. This allows us: to reduce the worst case complexity of decoding (the error performance being fixed) or to improve the error performance (for a same complexity); to reduce the ratio between average complexity and worst case complexity; to achieve practically optimal decoding of rate-1/2 codes of lengths up to 128 (rate-1/2 codes are a traditional benchmark, for coding rates different from 1/2, the decoding is easier); to achieve near-optimal decoding of a rate-1/2 code of length 192, which could never be performed before.     

Simple MAP decoding of first-order Reed-Muller and Hamming codes

A maximum a posteriori (MAP) probability decoder of a block code minimizes the probability of error for each transmitted symbol separately. The standard way of implementing MAP decoding of a linear code is the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm, which is based on a trellis representation of the code. The complexity of the BCJR algorithm for the first-order Reed-Muller (RM-1) codes and Hamming codes is proportional to n/sup 2/, where n is the code's length. In this correspondence, we present new MAP decoding algorithms for binary and nonbinary RM-1 and Hamming codes. The proposed algorithms have complexities proportional to q/sup 2/n log/sub q/n, where q is the alphabet size. In particular, for the binary codes this yields complexity of order n log n.     

Markov types and minimax redundancy for Markov sources

Redundancy of universal codes for a class of sources determines by how much the actual code length exceeds the optimal code length. In the minimax scenario, one designs the best code for the worst source within the class. Such minimax redundancy comes in two flavors: average minimax or worst case minimax. We study the worst case minimax redundancy of universal block codes for Markovian sources of any order. We prove that the maximal minimax redundancy for Markov sources of order r is asymptotically equal to 1/2m/sup r/(m-1)log/sub 2/n+log/sub 2/A/sub m//sup r/-(lnlnm/sup 1/(m-1)/)/lnm+o(1), where n is the length of a source sequence, m is the size of the alphabet, and A/sub m//sup r/ is an explicit constant (e.g., we find that for a binary alphabet m=2 and Markov of order r=1 the constant A/sub 2//sup 1/=16/spl middot/G/spl ap/14.655449504 where G is the Catalan number). Unlike previous attempts, we view the redundancy problem as an asymptotic evaluation of certain sums over a set of matrices representing Markov types. The enumeration of Markov types is accomplished by reducing it to counting Eulerian paths in a multigraph. In particular, we propose exact and asymptotic formulas for the number of strings of a given Markov type. All of these findings are obtained by analytic and combinatorial tools of analysis of algorithms.     

Efficient decoding algorithms for generalized Reed-Muller codes

Previously, a class of generalized Reed-Muller (RM) codes has been suggested for use in orthogonal frequency-division multiplexing. These codes offer error correcting capability combined with substantially reduced peak-to mean power ratios. A number of approaches to decoding these codes have already been developed. Here, we present low complexity, suboptimal alternatives which are inspired by the classical Reed decoding algorithm for binary RM codes. We simulate these new algorithms along with the existing decoding algorithms using additive white Gaussian noise and two-path fading models for a particular choice of code. The simulations show that one of our new algorithms outperforms all existing suboptimal algorithms and offers performance that is within 0.5 dB of maximum-likelihood decoding, yet has complexity comparable to or lower than existing decoding approaches     

A new Reed-Solomon code decoding algorithm based on Newton'sinterpolation

A Reed-Solomon code decoding algorithm based on Newton's interpolation is presented. This algorithm has as main application fast generalized-minimum-distance decoding of Reed-Solomon codes. It uses a modified Berlekamp-Massey algorithm to perform all necessary generalized-minimum-distance decoding steps in only one run. With a time-domain form of the new decoder the overall asymptotic generalized-minimum-distance decoding complexity becomes O(dn), with n the length and d the distance of the code (including the calculation of all error locations and values). This asymptotic complexity is optimal. Other applications are the possibility of fast decoding of Reed-Solomon codes with adaptive redundancy and a general parallel decoding algorithm with zero delay