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In this paper we consider the problem of encoding data into repeat-free sequences in which sequences are imposed to contain any k-tuple at most once (for predefined k). First, the capacity of the repeat-free constraint are calculated. Then, an efficient algorithm, which uses two bits of redundancy, is presented to encode length-n sequences for k = 2 + 2 log (n). This algorithm is then improved to support any value of k of the form k = a log (n), for 1 < a, while its redundancy is o(n). We also calculate the capacity of repeat-free sequences when combined with local constraints which are given by a constrained system, and the capacity of multi-dimensional repeat-free codes. [ABSTRACT FROM AUTHOR]

Hamming codes are of primary concern in information theory and its applications. Despite a number of researches that have been conducted on such codes and their characterizations, dealing with the properties of previously introduced codes in a BCK-algebraic framework has not been considered in earlier works. This paper investigates a code constructed based on BCK-algebraic models and proposes an algorithm corresponding to the presented code. It is noticeable that the suggested rendered algorithm is also established on the basis of the elements of a BCK-algebra. In fact, both the Hamming distance and dimension, associated with the presented code, can be estimated through a BCK-algebra structure due to the mechanism of its algorithm which is heavily dependent on the parity check matrix. In addition, the way in which the codes are designed contributes substantially to classification of them and to extract greater number of their attributes compared to the previous works. The highlight of the proposed method is that the number of atoms of the BCK-algebra plays a key role in calculation of the Hamming distance and dimension of these codes. Moreover, the obtained codes possess specified and recognizable Hamming distance which are essential in performing error-correcting, error-detecting and decoding tasks. [ABSTRACT FROM AUTHOR]

Recently, there were several papers on finding good linear codes. Various methods were introduced to make good linear codes. In this paper, we give a simple iterative algorithm to make good linear codes. Using our algorithm we construct 86 codes which improve the minimum distance of previously best known linear codes for fixed parameters n, k. [ABSTRACT FROM AUTHOR]

Reed Solomon (RS) codes are widely used to protect information from errors in transmission and storage systems. Most of the RS codes are based on GF(28) Galois Fields and use a byte to encode a symbol providing codewords up to 255 symbols. Codewords with more than 255 symbols can be obtained by using GF(2m) Galois Fields with m > 8, but this choice increases the complexity of the encoding and decoding algorithms. This limitation can be superseded by introducing Parity Sharing (PS) RS codes that are characterized by a greater flexibility in terms of design parameters. Consequently, a designer can choose between different PS code implementations in order to meet requirements such as Bit Error Rate (BER), hardware complexity, speed, and throughput. This paper analyzes the performance of PS codes in terms of BER with respect to the code parameters, taking into account either random error or erasure rates as two independent probabilities. This approach provides an evaluation that is independent of the communication channel characteristics and extends the results to memory systems in which permanent faults and transient faults can be modeled, respectively, as erasures and random errors. The paper also provides hardware implementations of the PS encoder and decoder and discusses their performances in terms of hardware complexity, speed, and throughput. [ABSTRACT FROM AUTHOR]

In this paper, we adopt a restricted Gaussian elimination on the Hankel structured augmented syndrome matrix to reinterpret an early-stopped version of the Berlekamp-Massey algorithm in which only (t + e) iterations are needed to be performed for the decoding of BCH codes up to t errors, where a is the number of errors actually occurred with e ≤ t, instead of the 2t iterations required in the conventional Berlekamp-Massey al- gorithm. The minimality of (t + e) iterations in this early-stopped Berlekamp-Massey (ESBM) algorithm is justified and related to the subject of simultaneous error correction and detection in this paper. We show that the multiplicative complexity of the ESBM algorithm is upper bounded by (te + e² - 1)∀e ≤ t and except for a trivial case, the ESBM algorithm is the most efficient algorithm for finding the error-locator polynomial. [ABSTRACT FROM AUTHOR]

This paper is motivated by the problem of error control in network coding when errors are introduced in a random fashion (rather than chosen by an adversary). An additive-multiplicative matrix channel is considered as a model for random network coding. The model assumes that n packets of length m are transmitted over the network, and up to t erroneous packets are randomly chosen and injected into the network. Upper and lower bounds on capacity are obtained for any channel parameters, and asymptotic expressions are provided in the limit of large field or matrix size. A simple coding scheme is presented that achieves capacity in both limiting cases. The scheme has decoding complexity O(n²m) and a probability of error that decreases exponentially both in the packet length and in the field size in bits. Extensions of these results for coherent network coding are also presented. [ABSTRACT FROM AUTHOR]

One way to find near-matches in large datasets is to use hash functions. In recent years locality-sensitive hash functions for various metrics have been given; for the Hamming metric projecting onto k bits is simple hash function that performs well. In this paper, we investigate alternatives to projection. For various parameters hash functions given by complete decoding algorithms for error-correcting codes work better, and asymptotically random codes perform better than projection. [ABSTRACT FROM AUTHOR]

In this paper, we present a practical approach to the iterative decoding of Reed-Solomon (RS) codes. The presented methodology utilizes an architecture in which the output produced by steps of belief-propagation (BP) is successively applied to a legacy decoding algorithm. Due to the suboptimal performance of BP conducted on the inherently dense RS parity-check matrix, a method is first provided for the construction of reduced-density, binary, parity-check equations. Iterative decoding is then conducted utilizing a subset of a redundant set of parity-check equations to minimize the number of connections into the least-reliable bits. Simulation results show that performance comparable to (and exceeding) the best known practical RS decoding techniques is achievable with the presented methodology. The complexity of the proposed algorithm is significantly lower than these existing procedures and permits a practical implementation in hardware. [ABSTRACT FROM AUTHOR]

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 ±1 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. [ABSTRACT FROM AUTHOR]

In this paper, we prove the following two results that expose some combinatorial limitations to list decoding Given n distinct elements α1,…,αn, from a field F, and n subsets S1,…Sn of F each of size at most l, the list decoding algorithm of Guruswami and Sudan can in polynomial time output all polynomials p of degree at most k that satisfy p(αi] ϵ Si for every i, as long as l ≤ [n⁄k]. We show that the performance of this algorithm is the best possible in a strong sense; specifically, when l = [n⁄k], the list of output polynomials can be superpolynomially large in n. For Reed-Solomon codes of block length n and dimension k + 1 where k = nδ for small enough δ, we exhibit an explicit received word with a superpolynomial number of Reed-Solomon codewords that agree with it on (2 - ϵ)k locations, for any desired ϵ ≥ 0 (agreement of k is trivial to achieve). Such a bound was known earlier only for a nonexplicit center. Finding explicit bad list decoding configurations is of significant interest—for example, the best known rate versus distance tradeoff, due to Xing, is based on a bad list decoding configuration for algebraic-geometric codes, which is unfortunately not explicitly known. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the joint estimation of the carrier frequency offset and statistical parameters of the multipath mobile channel. By modeling the multipath fading channel with a complex bandpass autoregressive (AR) model, we show how the estimates for the frequency offset, Doppler spread, and power profile of the multipath channel can be extracted from the estimated complex AR coefficients obtained via the expectation-maximization algorithm. A main advantage of the proposed joint estimator is that while it has a capability of performing equally well in all scattering environments, it can provide accurate estimates even in high-mobility channel conditions. We also demonstrate how the complexity of the estimator can be significantly reduced, while only slightly trading off performance, by applying the mean field approximation technique. Moreover, we derive a fully adaptive joint synchronization and channel-estimation scheme, as well as a novel Kalman-smoother-based frequency-error detector that can be used in feedback frequency-recovery schemes and is particularly well suited for fast-fading channel conditions. Finally, we revisit the Cramer-Rao lower bound analysis, and show how the Fisher information matrix can be conveniently computed in the presence of a frequency-selective Rayleigh fading channel. [ABSTRACT FROM AUTHOR]

Maximum-likelihood decoding is one of the central algorithmic problems in coding theory. It has been known for over 25 years that maximum-likelihood decoding of general linear codes is NP-hard. Nevertheless, it was so far unknown whether maximum-likelihood decoding remains hard for any specific family of codes with nontrivial algebraic structure. In this paper, we prove that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes. We moreover show that maximum-likelihood decoding of Reed-Solomon codes remains hard even with unlimited preprocessing, thereby strengthening a result of Bruck and Naor. [ABSTRACT FROM AUTHOR]

In this paper, a new decoding algorithm is proposed in which high-dimensional coding is done in a parity check code (high-dimensional ring code). The decoding method for the previously reported ring code has a complex correcting algorithm and requires much calculation time. We have focused on the point that a high-dimensional ring code can be divided into a number of two-dimensional ring codes, and by repeating the error correction of two-dimensional ring codes that requires small computation, we can perform error correction of high-dimensional ring codes. This decoding algorithm has a correction ability similar to the conventional decoding algorithms but with less computation. Moreover, when the error rate is high, random error and burst errors are mixed and an error correcting code is needed. However, since the decoding algorithm of the proposed code has a provision for dimensional division and the error generated on the transmitted block can be uniformly distributed on each two-dimensional plane, the error on the channel in the two-dimensional plane becomes random and error correction can be done efficiently. Moreover, if analysis or simulation increases the number of dimensions, then the correction ability is increased and the limit of the correction ability is determined from the size m of the code. The performance of convolutional codes and Reed Solomon codes is compared and it is shown that the ring code has high processing gain of error correction and that if the threshold point of correction lies in the high-error-rate region, then the decoding error rate is small and this code can be applied to high-error-rate correction. © 2001 Scripta Technica, Electron Comm Jpn Pt 3, 84(5): 75–85, 2001 [ABSTRACT FROM AUTHOR]