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Information Theory, IEEE Transactions on

Issue 1 • Date Jan. 2014

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Displaying Results 1 - 25 of 56
  • Table of contents

    Page(s): C1 - C4
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  • IEEE Transactions on Information Theory publication information

    Page(s): C2
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  • 2013 IEEE Information Theory Society Paper Award

    Page(s): 1 - 2
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  • 2013 IEEE Communications Society and Information Theory Society Joint Paper Award

    Page(s): 3 - 4
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  • Empirical Distribution of Good Channel Codes With Nonvanishing Error Probability

    Page(s): 5 - 21
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (4593 KB) |  | HTML iconHTML  

    This paper studies several properties of channel codes that approach the fundamental limits of a given (discrete or Gaussian) memoryless channel with a nonvanishing probability of error. The output distribution induced by an ϵ-capacity-achieving code is shown to be close in a strong sense to the capacity achieving output distribution. Relying on the concentration of measure (isoperimetry) property enjoyed by the latter, it is shown that regular (Lipschitz) functions of channel outputs can be precisely estimated and turn out to be essentially nonrandom and independent of the actual code. It is also shown that the output distribution of a good code and the capacity achieving one cannot be distinguished with exponential reliability. The random process produced at the output of the channel is shown to satisfy the asymptotic equipartition property. View full abstract»

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  • On Marton's Inner Bound and Its Optimality for Classes of Product Broadcast Channels

    Page(s): 22 - 41
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (8659 KB) |  | HTML iconHTML  

    Marton's inner bound is the tightest known inner bound on the capacity region of the broadcast channel. It is not known, however, if this bound is tight in general. One approach to settle this key open problem in network information theory is to investigate the multiletter extension of Marton's bound, which is known to be tight in general. This approach has become feasible only recently through the development of a new method for bounding cardinalities of auxiliary random variables by Gohari and Anantharam. This paper undertakes this long overdue approach to establish several new results, including 1) establishing the optimality of Marton's bound for new classes of product broadcast channels, 2) showing that the best-known outer bound by Nair and El Gamal is not tight in general, and 3) finding sufficient conditions for a global maximizer of Marton's bound that imply that the 2-letter extension does not increase the achievable rate. Motivated by the new capacity results, we establish a new outer bound on the capacity region of product broadcast channels. View full abstract»

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  • The Capacity Region of the Class of Three-Receiver Gaussian MIMO Multilevel Broadcast Channels With Two-Degraded Message Sets

    Page(s): 42 - 53
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (3542 KB) |  | HTML iconHTML  

    Nair and El Gamal established the capacity region of the three-receiver multilevel broadcast channel (MBC) with two-degraded message sets. For the three-receiver MBC with two-degraded message sets, the output at receiver 2 is a degraded version of the output at receiver 1. However, no order of degradedness is imposed on the output at receiver 3. The transmitter sends a common message to all three receivers and a private message to receiver 1. By considering a specific discrete-memoryless example, Nair and El Gamal showed that a direct extension of the Körner-Marton region (for the general two-receiver broadcast channel with degraded message sets) is strictly suboptimal. They also considered a three-receiver Gaussian product MBC and showed that, restricted to Gaussian inputs, the direct extension of the Körner-Marton region is again strictly suboptimal. However, whether Gaussian inputs are optimal remained unresolved. In this paper, we show that Gaussian inputs, along with time-sharing between rate points obtained with Gaussian inputs, achieve the capacity region of the three-receiver Gaussian multiple-input multiple-output MBC (this includes the three-receiver Gaussian product MBC considered by Nair and El Gamal) with two-degraded message sets. Our proof relies on the channel enhancement technique introduced by Weingarten as well as the perturbation approach employed by Liu and Viswanath. View full abstract»

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  • Coding Schemes and Asymptotic Capacity for the Gaussian Broadcast and Interference Channels With Feedback

    Page(s): 54 - 71
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (538 KB) |  | HTML iconHTML  

    A coding scheme is proposed for the memoryless Gaussian broadcast channel with correlated noises and feedback. For all noise correlations other than ±1, the gap between the sum-rate that the scheme achieves and the full-cooperation bound vanishes as the signal-to-noise ratio tends to infinity. When the correlation coefficient is -1, the gains afforded by feedback are unbounded and the prelog is doubled. When the correlation coefficient is +1, we demonstrate a dichotomy that if the noise variances are equal, then feedback is useless, and otherwise, feedback affords unbounded rate gains and doubles the prelog. The unbounded feedback gains, however, require perfect (noiseless) feedback. When the feedback links are noisy, the feedback gains are bounded, unless the feedback noise decays to zero sufficiently fast with the signal-to-noise ratio. Extensions to more receivers are also discussed as is the memoryless Gaussian interference channel with feedback. View full abstract»

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  • Analytical and Numerical Characterizations of Shannon Ordering for Discrete Memoryless Channels

    Page(s): 72 - 83
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (3180 KB) |  | HTML iconHTML  

    This paper studies several problems concerning channel inclusion, which is a partial ordering between discrete memoryless channels (DMCs) proposed by Shannon. Specifically, majorization-based conditions are derived for channel inclusion between certain DMCs. Furthermore, under general conditions, channel equivalence defined through Shannon ordering is shown to be the same as permutation of input and output symbols. The determination of channel inclusion is considered as a convex optimization problem, and the sparsity of the weights related to the representation of the worse DMC in terms of the better one is revealed when channel inclusion holds between two DMCs. For the exploitation of this sparsity, an effective iterative algorithm is established based on modifying the orthogonal matching pursuit algorithm. View full abstract»

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  • Symmetrical Multilevel Diversity Coding and Subset Entropy Inequalities

    Page(s): 84 - 103
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (513 KB) |  | HTML iconHTML  

    Symmetrical multilevel diversity coding (SMDC) is a classical model for coding over distributed storage. In this setting, a simple separate encoding strategy known as superposition coding was shown to be optimal in terms of achieving the minimum sum rate and the entire admissible rate region of the problem. The proofs utilized carefully constructed induction arguments, for which the classical subset entropy inequality played a key role. This paper consists of two parts. In the first part, the existing optimality proofs for classical SMDC are revisited, with a focus on their connections to subset entropy inequalities. Initially, a new sliding-window subset entropy inequality is introduced and then used to establish the optimality of superposition coding for achieving the minimum sum rate under a weaker source-reconstruction requirement. Finally, a subset entropy inequality recently proved by Madiman and Tetali is used to develop a new structural understanding of the work of Yeung and Zhang on the optimality of superposition coding for achieving the entire admissible rate region. Building on the connections between classical SMDC and the subset entropy inequalities developed in the first part, in the second part the optimality of superposition coding is extended to the cases where there is either an additional all-access encoder or an additional secrecy constraint. View full abstract»

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  • Sharp Inequalities for f -Divergences

    Page(s): 104 - 121
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (722 KB) |  | HTML iconHTML  

    f-divergences are a general class of divergences between probability measures which include as special cases many commonly used divergences in probability, mathematical statistics, and information theory such as Kullback-Leibler divergence, chi-squared divergence, squared Hellinger distance, total variation distance, and so on. In this paper, we study the problem of maximizing or minimizing an f-divergence between two probability measures subject to a finite number of constraints on other f-divergences. We show that these infinite-dimensional optimization problems can all be reduced to optimization problems over small finite dimensional spaces which are tractable. Our results lead to a comprehensive and unified treatment of the problem of obtaining sharp inequalities between f-divergences. We demonstrate that many of the existing results on inequalities between f-divergences can be obtained as special cases of our results. We also improve on some existing non-sharp inequalities. View full abstract»

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  • Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-Rank Matrices

    Page(s): 122 - 132
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (341 KB) |  | HTML iconHTML  

    This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key technical tool, which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while yielding sharp results. It is shown that for any given constant t ≥ 4/3, in compressed sensing, δtkA <; √((t-1)/t) guarantees the exact recovery of all k sparse signals in the noiseless case through the constrained l1 minimization, and similarly, in affine rank minimization, δtrM <; √((t-1)/t) ensures the exact reconstruction of all matrices with rank at most r in the noiseless case via the constrained nuclear norm minimization. In addition, for any ε > 0, δtkA <; √(t-1/t) + ε is not sufficient to guarantee the exact recovery of all k-sparse signals for large k. Similar results also hold for matrix recovery. In addition, the conditions δtkA <; √((t-)1/t) and δtrM <; √((t-1)/t) are also shown to be sufficient, respectively, for stable recovery of approximately sparse signals and low-rank matrices in the noisy case. View full abstract»

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  • On the Fundamental Limits of Recovering Tree Sparse Vectors From Noisy Linear Measurements

    Page(s): 133 - 149
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (562 KB) |  | HTML iconHTML  

    Recent breakthrough results in compressive sensing (CS) have established that many high dimensional signals can be accurately recovered from a relatively small number of non-adaptive linear observations, provided that the signals possess a sparse representation in some basis. Subsequent efforts have shown that the performance of CS can be improved by exploiting additional structure in the locations of the nonzero signal coefficients during inference or by utilizing some form of data-dependent adaptive measurement focusing during the sensing process. To the best of our knowledge, our own previous work was the first to establish the potential benefits that can be achieved when fusing the notions of adaptive sensing and structured sparsity. In that work, we examined the task of support recovery from noisy linear measurements, and established that an adaptive sensing strategy specifically tailored to signals that are tree-sparse can significantly outperform adaptive and non-adaptive sensing strategies that are agnostic to the underlying structure. In this paper, we establish fundamental performance limits for the task of support recovery of tree-sparse signals from noisy measurements, in settings where measurements may be obtained either non-adaptively (using a randomized Gaussian measurement strategy motivated by initial CS investigations) or by any adaptive sensing strategy. Our main results here imply that the adaptive tree sensing procedure analyzed in our previous work is nearly optimal, in the sense that no other sensing and estimation strategy can perform fundamentally better for identifying the support of tree-sparse signals. View full abstract»

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  • Signal Estimation With Additive Error Metrics in Compressed Sensing

    Page(s): 150 - 158
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (2194 KB) |  | HTML iconHTML  

    Compressed sensing typically deals with the estimation of a system input from its noise-corrupted linear measurements, where the number of measurements is smaller than the number of input components. The performance of the estimation process is usually quantified by some standard error metric such as squared error or support set error. In this correspondence, we consider a noisy compressed sensing problem with any additive error metric. Under the assumption that the relaxed belief propagation method matches Tanaka's fixed point equation, we propose a general algorithm that estimates the original signal by minimizing the additive error metric defined by the user. The algorithm is a pointwise estimation process, and thus simple and fast. We verify that our algorithm is asymptotically optimal, and we describe a general method to compute the fundamental information-theoretic performance limit for any additive error metric. We provide several example metrics, and give the theoretical performance limits for these cases. Experimental results show that our algorithm outperforms methods such as relaxed belief propagation (relaxed BP) and compressive sampling matching pursuit (CoSaMP), and reaches the suggested theoretical limits for our example metrics. View full abstract»

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  • Discretized Gabor Frames of Totally Positive Functions

    Page(s): 159 - 169
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (407 KB) |  | HTML iconHTML  

    In this paper, a large class of universal windows for Gabor frames (Weyl-Heisenberg frames) is constructed. These windows have the fundamental property that every overcritical rectangular lattice generates a Gabor frame. Likewise, every undercritical rectangular lattice generates a Riesz sequence. View full abstract»

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  • Kirkman Equiangular Tight Frames and Codes

    Page(s): 170 - 181
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (2988 KB) |  | HTML iconHTML  

    An equiangular tight frame (ETF) is a set of unit vectors in a Euclidean space whose coherence is as small as possible, equaling the Welch bound. Also known as Welch-bound-equality sequences, such frames arise in various applications, such as waveform design and compressed sensing. At the moment, there are only two known flexible methods for constructing ETFs: harmonic ETFs are formed by carefully extracting rows from a discrete Fourier transform; Steiner ETFs arise from a tensor-like combination of a combinatorial design and a regular simplex. These two classes seem very different: the vectors in harmonic ETFs have constant amplitude, whereas Steiner ETFs are extremely sparse. We show that they are actually intimately connected: a large class of Steiner ETFs can be unitarily transformed into constant-amplitude frames, dubbed Kirkman ETFs. Moreover, we show that an important class of harmonic ETFs is a subset of an important class of Kirkman ETFs. This connection informs the discussion of both types of frames: some Steiner ETFs can be transformed into constant-amplitude waveforms making them more useful in waveform design; some harmonic ETFs have low spark, making them less desirable for compressed sensing. We conclude by showing that real-valued constant-amplitude ETFs are equivalent to binary codes that achieve the Grey-Rankin bound, and then construct such codes using Kirkman ETFs. View full abstract»

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  • Burst List Decoding of Interleaved Reed–Solomon Codes

    Page(s): 182 - 190
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (518 KB) |  | HTML iconHTML  

    It is shown that interleaved Reed-Solomon codes can be list-decoded for burst errors while attaining the generalized Reiger bound for list decoding. A respective decoding algorithm is presented that is (significantly) more efficient than a burst list decoder for a noninterleaved Reed-Solomon code with comparable parameters. Finally, it is shown through counterexamples that unlike the special case of Reed-Solomon codes, interleaving does not always preserve the list decoding properties of the constituent code. View full abstract»

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  • On Decoding Irregular Tanner Codes With Local-Optimality Guarantees

    Page(s): 191 - 211
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (5634 KB) |  | HTML iconHTML  

    We consider decoding of binary linear Tanner codes using message-passing iterative decoding and linear-programming (LP) decoding in memoryless binary-input output-symmetric (MBIOS) channels. We present new certificates that are based on a combinatorial characterization for the local optimality of a codeword in irregular Tanner codes with respect to any MBIOS channel. This characterization is a generalization of (Arora , Proc. ACM Symp. Theory of Computing, 2009) and (Vontobel, Proc. Inf. Theory and Appl. Workshop, 2010) and is based on a conical combination of normalized weighted subtrees in the computation trees of the Tanner graph. These subtrees may have any finite height h (even equal or greater than half of the girth of the Tanner graph). In addition, the degrees of local-code nodes in these subtrees are not restricted to two (i.e., these subtrees are not restricted to skinny trees). We prove that local optimality in this new characterization implies maximum-likelihood (ML) optimality and LP optimality, and show that a certificate can be computed efficiently. We also present a new message-passing iterative decoding algorithm, called normalized weighted min-sum (NWMS). NWMS decoding is a belief-propagation (BP) type algorithm that applies to any irregular binary Tanner code with single parity-check local codes (e.g., low-density and high-density parity-check codes). We prove that if a locally optimal codeword with respect to height parameter h exists (whereby notably h is not limited by the girth of the Tanner graph), then NWMS decoding finds this codeword in h iterations. The decoding guarantee of the NWMS decoding algorithm applies whenever there exists a locally optimal codeword. Because local optimality of a codeword implies that it is the unique ML codeword, the decoding guarantee also provides an ML certificate for this codeword. Finally, we apply the new local-optimality characterization to regular Tanner codes, and prove lower bounds on the noise thresh- lds of LP decoding in MBIOS channels. When the noise is below these lower bounds, the probability that LP decoding fails to decode the transmitted codeword decays doubly exponentially in the girth of the Tanner graph. View full abstract»

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  • Optimal Locally Repairable and Secure Codes for Distributed Storage Systems

    Page(s): 212 - 236
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1118 KB) |  | HTML iconHTML  

    This paper aims to go beyond resilience into the study of security and local-repairability for distributed storage systems (DSSs). Security and local-repairability are both important as features of an efficient storage system, and this paper aims to understand the trade-offs between resilience, security, and local-repairability in these systems. In particular, this paper first investigates security in the presence of colluding eavesdroppers, where eavesdroppers are assumed to work together in decoding the stored information. Second, this paper focuses on coding schemes that enable optimal local repairs. It further brings these two concepts together to develop locally repairable coding schemes for DSS that are secure against eavesdroppers. The main results of this paper include: 1) an improved bound on the secrecy capacity for minimum storage regenerating codes; 2) secure coding schemes that achieve the bound for some special cases; 3) a new bound on minimum distance for locally repairable codes; 4) code construction for locally repairable codes that attain the minimum distance bound; and 5) repair-bandwidth-efficient locally repairable codes with and without security constraints. View full abstract»

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  • A Contention-Free Parallel Access by Butterfly Networks for Turbo Interleavers

    Page(s): 237 - 251
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1189 KB) |  | HTML iconHTML  

    A theoretical foundation for any turbo interleaver to be a contention-free interleaver to access data in parallel by a butterfly network is presented. A contention-free parallel access of multiple memories in parallel plays a crucial role for implementing high speed turbo decoders for high data rate applications. The presented theoretical analysis shows that a butterfly network has a sufficiently rich permutation structure to be a routing network between parallel decoder units and multiple memories. Thus turbo code design is independent of the designing of a contention-free parallel access by butterfly networks. In particular, a turbo interleaver needs not to provide a built-in contention-free parallel access for any parallel access by butterfly networks. We demonstrate how to apply this theory to turbo interleavers widely used in commercial telecommunication standards. View full abstract»

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  • G -Equivalence in Group Algebras and Minimal Abelian Codes

    Page(s): 252 - 260
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (2994 KB) |  | HTML iconHTML  

    Let G be a finite Abelian group and BBF a field such that char(BBF ) does not divide |G|. Denote by BBF G the group algebra of G over BBF. A (semisimple) Abelian code is an ideal of BBF G. Two codes ℑ1 and ℑ2 of BBF G are G-equivalent if there exists an automorphism ψ of G whose linear extension to BBF G maps ℑ1 onto ℑ2. In this paper, we give a necessary and sufficient condition for minimal Abelian codes to be G-equivalent and show how to correct some results in the literature. View full abstract»

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  • On the Error Detection Capability of One Check Digit

    Page(s): 261 - 270
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (289 KB) |  | HTML iconHTML  

    In this paper, we study a check digit system which is based on the use of elementary abelian p-groups of order pk. This paper is inspired by a recently introduced check digit system for hexadecimal numbers. By interpreting its check equation in terminology of matrix algebra, we generalize the idea to build systems over a group of order pk, while keeping the ability to detect all the: 1) single errors; 2) adjacent transpositions; 3) twin errors; 4) jump transpositions; and 5) jump twin errors. Besides, we consider two categories of jump errors: 1) t-jump transpositions and 2) t-jump twin errors, which include and further extend the double error types of 2)-5). In particular, we explore Rc, the maximum detection radius of the system on detecting these two kinds of generalized jump errors, and show that it is 2k-2 for p=2 and (pk-1)/2-1 for an odd prime p. Also, we show how to build such a system that detects all the single errors and these two kinds of double jump-errors within Rc. View full abstract»

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  • Gray Codes and Enumerative Coding for Vector Spaces

    Page(s): 271 - 281
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (2769 KB) |  | HTML iconHTML  

    Gray codes for vector spaces are considered in two graphs: the Grassmann graph, and the projective-space graph, both of which have recently found applications in network coding. For the Grassmann graph, constructions of cyclic optimal codes are given for all parameters. As for the projective-space graph, two constructions for specific parameters are provided, as well some nonexistence results. Furthermore, encoding and decoding algorithms are given for the Grassmannian Gray code, which induce an enumerative-coding scheme. The computational complexity of the algorithms is at least as low as known schemes, and for certain parameter ranges, the new scheme outperforms previously known ones. View full abstract»

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  • On the Weight Hierarchy of Codes Coming From Semigroups With Two Generators

    Page(s): 282 - 295
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (5072 KB) |  | HTML iconHTML  

    The weight hierarchy of one-point algebraic geometry codes can be estimated by means of the generalized order bounds, which are described in terms of a certain Weierstrass semigroup. The asymptotical behavior of such bounds for r ≥ 2 differs from that of the classical Feng-Rao distance (r=1) by the so-called Feng-Rao numbers. This paper is addressed to compute the Feng-Rao numbers for numerical semigroups of embedding dimension two (with two generators), obtaining a closed simple formula for the general case by using numerical semigroup techniques. These involve the computation of the Apéry set with respect to an integer of the semigroups under consideration. The formula obtained is applied to lower bounding the generalized Hamming weights, improving the bound given by Kirfel and Pellikaan in terms of the classical Feng-Rao distance. We also compare our bound with a modification of the Griesmer bound, improving this one in many cases. View full abstract»

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  • Weight Distributions of Two Classes of Cyclic Codes With Respect to Two Distinct Order Elements

    Page(s): 296 - 303
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (647 KB) |  | HTML iconHTML  

    Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Cyclic codes have been studied for many years, but their weight distributions are known only for a few cases. In this paper, let Fr be an extension of a finite field Fq and r = qm, we determine the weight distributions of the cyclic codes C={c(a, b): a, b ∈ Fr}, c(a, b)= Trr/q(ag10 +bg20),...,Trr/q(ag1n-1+bg2n-1)), g1, g2 ∈ Fr, in the following two cases: 1) ord(g1)=n, n|r-1 and g2=1 and 2) ord(g1)=n, g2=g12, ord(g2)=n/2, m=2, and 2(r-1)/n|(q+1). View full abstract»

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Aims & Scope

IEEE Transactions on Information Theory publishes papers concerned with the transmission, processing, and utilization of information.

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Meet Our Editors

Editor-in-Chief
Frank R. Kschischang

Department of Electrical and Computer Engineering