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Cyclic codes have been widely used in many applications of communication systems and data storage systems. This paper proposes a new procedure for decoding cyclic codes up to actual minimum distance. The decoding procedure consists of two steps: 1) computation of known syndromes and 2) computation of error positions and error values simultaneously. To do so, a matrix whose all entries are syndromes is called syndrome matrix. A matrix whose entries are either syndromes or the elements of a finite field is said to be partial syndrome matrix. In this paper, two novel methods are presented to determine error positions and error values simultaneously and directly. The first method uses a new partial syndrome matrix along with Gaussian elimination. The partial syndrome matrices for binary (respectively, ternary) cyclic codes of lengths from 69 to 99 (respectively, 16 to 37) are tabulated. For some cyclic codes, the partial syndrome matrices contain unknown syndromes; the second method constructs a matrix from a system of equations, which is generated by the determinants of different partial syndrome matrices and makes use of Gaussian elimination to determine its row rank. Many more cyclic codes beyond the Bose–Chaudhuri–Hocquenghem bound can be decoded with these methods. [ABSTRACT FROM AUTHOR]

Cyclic codes are widely employed in communication systems, storage devices, and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over \mathrm GF(q) with length n=(q^m-1)/(q-1) . Little is known about this class of BCH codes when $q>2$ . The objective of this paper is to study some of the codes within this class. In particular, the dimension, the minimum distance, and the weight distribution of some ternary BCH codes with length n=(3^{m}-1)/2 are determined in this paper. A class of ternary BCH codes meeting the Griesmer bound is identified. An application of some of the BCH codes in secret sharing is also investigated. [ABSTRACT FROM PUBLISHER]

Let $q$ be a prime power and $m\geq 3$ be odd. Suppose that $n=\frac {q^{2m}-1}{a}$ with $a|(q^{m}+1)$ and $3\leq a \leq 2(q^{2}-q+1)$. This paper mainly determines the actual maximum designed distance of Hermitian dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes over $\mathbb {F}_{q^{2}}$ of length $n$. Firstly, we give the maximum designed distance $\delta _{m,a}^{R}$ of narrow-sense Hermitian dual-containing BCH codes. Secondly, we show that there are also non-narrow-sense ones of designed distance up to $\delta _{m,a}^{R}$. It is worth mentioning that our maximum designed distance $\delta _{m,a}^{R}>\lceil \frac {a}{2}\rceil \delta _{m}^{A}$ , where $\delta _{m}^{A}$ is given by Aly et al. (IEEE Trans. Inf. Theory, vol. 53, no. 3, pp. 1183-1188, 2007). Thus, many families of Hermitian dual-containing BCH codes with relatively large designed distance are obtained. Using the Hermitian construction to them, we can subsequently construct different classes of nonprimitive quantum codes, which are new in the sense that their parameters are not covered in the literature. [ABSTRACT FROM AUTHOR]

The tone reservation method can be used to reduce the peak to average power ratio (PAPR) in orthogonal frequency division multiplexing (OFDM) transmission systems. In this paper, the tone reservation method is analyzed for OFDM with a restricted carrier set, where only the positive carrier frequencies are used. Performing a fully analytical analysis, we give a complete characterization of the information sets for which the PAPR problem is solvable. To derive our main result, we connect the PAPR problem with a geometric functional analytic property of certain spaces. Furthermore, we present applications of our theory that give guidelines for choosing the information carriers in the finite setting and discuss a probabilistic approach for selecting the carriers. In addition, it is shown that if there exists an information sequence for which the PAPR problem is not solvable, then the set of information sequences for which the PAPR problem is not solvable is a residual set. [ABSTRACT FROM AUTHOR]

In general, it is hard to find the cross correlation distribution between two sequences. In particular, there are only few cases that the cross correlation is three-valued. In this paper, new pairs of binary sequences with three cross correlation values are presented. These two sequences have different lengths: 2^2k-1 and 3(2^k-1) , respectively. We determine the cross correlation distribution between these two sequences. We also give the possible correlation values for the shift decimated sequences. The magnitude of cross correlation values is shown to be low. The techniques presented in this paper may be useful to handle other similar cases. [ABSTRACT FROM PUBLISHER]

In 1964, Massey introduced a class of codes with complementary duals which are called linear complimentary dual (LCD) codes. He showed that LCD codes have applications in communication system, side-channel attack and so on. LCD codes have been extensively studied in literature. On the other hand, MDS codes form an optimal family of classical codes which have wide applications in both theory and practice. The main purpose of this paper is to give an explicit construction of several classes of LCD MDS codes, using tools from algebraic function fields. We exemplify this construction and obtain several classes of explicit LCD MDS codes for the odd characteristic case. [ABSTRACT FROM AUTHOR]

This paper investigates the joint source-channel coding problem of sending a memoryless source over a memoryless broadcast channel. An inner bound and several outer bounds on the admissible distortion region are derived, which, respectively, generalize and unify several existing bounds. As a consequence, we also obtain an inner bound and an outer bound for the degraded broadcast channel case. When specialized to the Gaussian or binary source broadcast, the inner bound and outer bound not only recover the best known inner bound and outer bound in the literature but also generate some new results. Besides, we also extend the inner bound and outer bounds to the Wyner–Ziv source broadcast problem, i.e., source broadcast with side information available at decoders. Some new bounds are obtained when specialized to the Wyner–Ziv Gaussian and Wyner–Ziv binary cases. [ABSTRACT FROM AUTHOR]

Frequency-hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, lower bounds on the partial Hamming correlation of FHSs and FHS sets are proposed. They slightly improve the known bounds by Eun et al. and Zhou et al. A construction of FHSs and FHS sets having optimal partial Hamming correlation with respect to the improved bounds is also presented based on the theory of generalized cyclotomy. Our construction yields optimal FHSs and FHS sets with new and flexible parameters not covered in this paper. [ABSTRACT FROM AUTHOR]

Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for given length and code size. Constructing LCD MDS codes is thus of significance in theory and practice. Recently, Jin constructed several classes of LCD MDS codes through generalized Reed-Solomon codes. In this paper, a different approach is proposed to obtain new LCD MDS codes from generalized Reed-Solomon codes. Consequently, new code constructions are provided and certain previously known results by Jin are extended. [ABSTRACT FROM AUTHOR]

In this paper, we consider a lossy single-user caching problem with correlated sources. We first describe the fundamental interplay between the source correlations, the capacity of the user’s cache, the user’s reconstruction distortion requirements, and the final delivery-phase (compression) rate. We then illustrate this interplay using a multivariate Gaussian source example and a binary symmetric source example. To fully explore the effect of the user’s distortion requirements, we formulate the caching problem using \mathsf f -separable distortion functions recently introduce by Shkel and Verdú. The class of \mathsf f -separable distortion functions includes separable distortion functions as a special case, and our analysis covers both the expected- and excess-distortion settings in detail. We also determine what “common information” should be placed in the cache, and what information should be transmitted during the delivery phase. To this end, two new common-information measures are introduced for caching, and their relationship to the common-information measures of Wyner, Gács, and Körner is discussed in detail. [ABSTRACT FROM PUBLISHER]

We consider a multi-channel communication system in which a transmitter has access to $M$ channels, but does not know the state of any of the channels. We model the channel state using an ON/OFF Markov process, and allow the transmitter to probe a single channel at predetermined probing intervals to decide over which channel to transmit. For models in which the transmitter must transmit over the probed channel, it has been shown that a myopic policy probing the channel most likely to be ON is optimal. In this paper, we allow the transmitter to select a channel over which to transmit that is potentially different from the probed channel. For a system of two channels, we show that the choice of which channel to probe does not affect the throughput. For a system with many channels, we show that a probing policy that probes the channel that is the second-most likely to be ON results in higher throughput. We extend the channel probing problem to dynamically choose when to probe based on probing history, and characterize the optimal probing policy for various scenarios. [ABSTRACT FROM PUBLISHER]

The objective of low-delay codes is to protect communication streams from erasure bursts by minimizing the time between the packet erasure and its reconstruction. Previous work has concentrated on the constant-delay scenario, where all erased packets need to exhibit the same decoding delay. We consider the case of heterogeneous delay, where the objective is to minimize the average delay across the erased packets in a burst. We derive delay lower bounds for the average case, and show that they match the constant-delay bounds only at a single rate point $R=0.5$ . We then construct codes with optimal average delays for the entire range of code rates. The construction for rates $R\leq 0.5$ achieves optimality for every erasure instance, while the construction for rates $R>0.5$ is optimal for a $(1-R)/R$ fraction of all burst instances and close to optimal for the remaining fraction. The paper also studies the benefits of delay heterogeneity within the application of sensor communications. It is shown that a carefully designed code can significantly improve the temporal precision at the receiving node following erasure-burst events. [ABSTRACT FROM PUBLISHER]

We define the common randomness-assisted capacity of an arbitrarily varying wiretap channel (AVWC) when the eavesdropper is kept ignorant about the common randomness. We prove a multi-letter capacity formula for this model. We prove that, if enough common randomness is used, the capacity formula can be given a single-shot form again. We then consider the opposite extremal case, where no common randomness is available, and derive the capacity. It is known that the capacity of the system can be discontinuous under these circumstances. We prove here that it is still stable in the sense that it is continuous around its positivity points. We further prove that discontinuities can only arise if the legal link is symmetrizable and characterize the points where it is positive. These results shed new light on the design principles of communication systems with embedded security features. At last, we investigate the effect of super-activation of the message transmission capacity of AVWCs under the average error criterion. We give a complete characterization of those AVWCs that may be super-activated. The effect is thereby also related to the (conjectured) super-activation of the common randomness assisted capacity of AVWCs with an eavesdropper that gets to know the common randomness. Super-activation is based on the idea of wasting a few bits of non-secret messages in order to enable provably secret transmission of a large bulk of data, a concept that may prove to be of further importance in the design of communication systems. In this paper, we provide further insight into this phenomenon by providing a class of codes that is capacity achieving and does not convey any information to the eavesdropper. [ABSTRACT FROM AUTHOR]

Frequency hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, we investigate constructions of FHSs and FHS sets with optimal partial Hamming correlation. We first establish a correspondence between FHS sets with optimal partial Hamming correlation and multiple partition-type balanced nested cyclic difference packings with a special property. By virtue of this correspondence, some FHSs and FHS sets with optimal partial Hamming correlation are constructed from various combinatorial structures, such as cyclic difference packings and cyclic relative difference families. As a consequence, our constructions yield new FHSs and FHS sets with optimal partial Hamming correlation. [ABSTRACT FROM AUTHOR]

Frequency-hopping sequences (FHSs) with favorable partial Hamming correlation properties are desirable in many synchronization and multiple-access systems. An FHS set is said to be strictly optimal if it has optimal partial Hamming correlation for any correlation window. In this paper, we derive upper bounds on the family sizes of FHS sets with respect to partial Hamming correlation from some classical bounds on error-correcting codes. We then present strictly optimal FHS sets having optimal family sizes with respect to one of the new bounds. In particular, our construction gives new parameters not covered in the literature. [ABSTRACT FROM AUTHOR]