Explicit Code Constructions Research Articles

Exploring Node Repair Locality in Fractional Repetition Codes

In this letter, we focus on fractional repetition (FR) codes, which enable uncoded exact node repair with the minimum disk reads and network bandwidth. We investigate the problem of designing FR codes that also incorporate the preferable property of local repair , i.e., the nodes participating in the repair operation are less than those required for data reconstruction. We propose explicit code constructions based on the combinatorial structure called symmetric design . The benefits of the proposed construction include: 1) each contacted helper node transfers the same amount of data in the repair process, and 2) each failed storage node can have multiple local repair alternatives.

Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs

We construct an explicit family of $ \mathbb {F}_{h}$ -linear rank-metric codes over any field $ \mathbb {F}_{h}$ that enables efficient list-decoding up to a fraction $\rho $ of errors in the rank metric with a rate of $1-\rho - \varepsilon $ , for any desired $\rho \in (0,1)$ and $ \varepsilon > 0$ . This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an $ \mathbb {F}_{h}$ -subspace that evades the structured subspaces over an extension field $ \mathbb {F}_{h^{t}}$ that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS’13) with subspace-evasive varieties due to Dvir and Lovett (STOC’12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed–Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes.

Cooperative repair of multiple node failures in distributed storage systems

Cooperative regenerating codes are designed for repairing multiple node failures in distributed storage systems. In contrast to the original repair model of regenerating codes, which are for the repair of single node failure, data exchange among the new nodes is enabled. It is known that further reduction in repair bandwidth is possible with cooperative repair. Currently in the literature, we have an explicit construction of exact-repair cooperative code achieving all parameters corresponding to the minimum-bandwidth point. We give a slightly generalized and more flexible version of this cooperative regenerating code in this paper. For minimum-storage regeneration with cooperation, we present an explicit code construction which can jointly repair any number of systematic storage nodes.

Adaptive Fractional Repetition Codes for Dynamic Storage Systems

Node recovery is a challenging task in erasure-coded storage systems, which involves a considerable amount of network resources. The efficacy of failure recovery is measured by various metrics, such as repair bandwidth and computational complexity. This letter focuses on fractional repetition (FR) codes, which are a family of codes that provide uncoded exact repairs with minimum repair bandwidth. We consider the usage of FR codes in dynamic storage systems, where the system parameters can vary over time. Specifically, we introduce adaptive fractional repetition codes that can effectively adapt to system changes. The proposed codes allow update of systems without the need for frequent reconfigurations. Moreover, explicit code constructions for both homogeneous and heterogeneous storage systems are presented.

ONE-HOMOGENEOUS WEIGHT CODES OVER FINITE CHAIN RINGS

Abstract. This paper determines the structures of one-homogeneousweight codes over finite chain rings and studies the algebraic propertiesof these codes. We present explicit constructions of one-homogeneousweight codes over finite chain rings. By taking advantage of the distance-preserving Gray map defined in [7] from the finite chain ring to its residuefield, we obtain a family of optimal one-Hamming weight codes over theresidue field. Further, we propose a generalized method that also includesthe examples of optimal codes obtained by Shi et al. in [17]. 1. IntroductionConstant-weight codes represent an important class of codes within the fam-ily of error-correcting codes [11]. A linear code having constant-weight meansthat every nonzero codeword has the same weight. In the literature there aremany papers on binary constant-weight codes which have several applicationssuch as the design of demultiplexers for nano-scale memories [8] and the con-struction of frequency hopping lists for use in GSM networks [12]. Especially,considerable research has been done on the central problem regarding constant-weight codes which is the determination of A(n,d,w), the largest possible sizeof a constant-weight code of length n, Hamming distance at least d, and con-stant weight w. Due to the difficulty in finding good constant-weight codes,various upper and lower bounds on A(n,d,w) have been developed [1, 3, 15, 18].Moreover, there are further studies in this direction including nonbinary finitefields in [2, 10]. It has been shown that there exists a unique one-weight binarylinear code of dimension k such that any two columns in its generator matrixare linearly independent for every positive integer k. Later, this result has beenextended to the ring Z

An Error Event Sensitive Tradeoff Between Rate and Coding Gain in MIMO MAC

This paper investigates the design of codes for multiple-input multiple-output (MIMO) multiple access channel (MAC). If a joint maximum-likelihood decoding is to be performed at the receiver, then every MIMO-MAC code can be regarded as a single-user code, where the minimum determinant criterion proposed by Tarokh et al. is useful for designing such codes and for upper bounding the maximum pairwise error probability (PEP), whenever the codes are of finite rate and operate in finite signal-to-noise ratio range. Unlike the case of single-user codes where the minimum determinant can be lower bounded by a fixed constant as code-rate grows, it was proved by Lahtonen et al. that the minimum determinant of MIMO-MAC codes decays as a function of the rates. This decay phenomenon is further investigated in this paper, and upper bounds for the decays of minimum determinant corresponding to each error event are provided. Lower bounds for the optimal decay are established and are based on an explicit construction of codes using algebraic number theory and Diophantine approximation. For some error profiles, the constructed codes are shown to meet the aforementioned upper bounds, hence they are optimal finite-rate codes in terms of PEPs associated with such error events. An asymptotic diversity-multiplexing gain tradeoff (DMT) analysis of the proposed codes is also given. It is shown that these codes are DMT optimal when the values of multiplexing gains are small.

HFR code: a flexible replication scheme for cloud storage systems

Fractional repetition (FR) codes are a family of repair-efficient storage codes that provide exact and uncoded node repair at the minimum bandwidth regenerating point. The advantageous repair properties are achieved by a tailor-made two-layer encoding scheme which concatenates an outer maximum-distance-separable (MDS) code and an inner repetition code. In this paper, we generalize the application of FR codes and propose heterogeneous fractional repetition (HFR) code, which is adaptable to the scenario where the repetition degrees of coded packets are different. We provide explicit code constructions by utilizing group divisible designs, which allow the design of HFR codes over a large range of parameters. The constructed codes achieve the system storage capacity under random access repair and have multiple repair alternatives for node failures. Further, we take advantage of the systematic feature of MDS codes and present a novel design framework of HFR codes, in which storage nodes can be wisely partitioned into clusters such that data reconstruction time can be reduced when contacting nodes in the same cluster.

Non-malleable Coding Against Bit-Wise and Split-State Tampering

Non-malleable coding, introduced by Dziembowski et al. (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Non-malleable coding is possible against any class of adversaries of bounded size. In particular, Dziembowski et al. show that such codes exist and may achieve positive rates for any class of tampering functions of size at most $$2^{2^{\alpha n}}$$22źn, for any constant $$\alpha \in [0, 1)$$źź[0,1). However, this result is existential and has thus attracted a great deal of subsequent research on explicit constructions of non-malleable codes against natural classes of adversaries. In this work, we consider constructions of coding schemes against two well-studied classes of tampering functions; namely, bit-wise tampering functions (where the adversary tampers each bit of the encoding independently) and the much more general class of split-state adversaries (where two independent adversaries arbitrarily tamper each half of the encoded sequence). We obtain the following results for these models. (1) For bit-tampering adversaries, we obtain explicit and efficiently encodable and decodable non-malleable codes of length n achieving rate $$1-o(1)$$1-o(1) and error (also known as exact security) $$\exp (-\tilde{\varOmega }(n^{1/7}))$$exp(-Ω~(n1/7)). Alternatively, it is possible to improve the error to $$\exp (-\tilde{\varOmega }(n))$$exp(-Ω~(n)) at the cost of making the construction Monte Carlo with success probability $$1-\exp (-\varOmega (n))$$1-exp(-Ω(n)) (while still allowing a compact description of the code). Previously, the best known construction of bit-tampering coding schemes was due to Dziembowski et al. (ICS 2010), which is a Monte Carlo construction achieving rate close to .1887. (2) We initiate the study of seedless non-malleable extractors as a natural variation of the notion of non-malleable extractors introduced by Dodis and Wichs (STOC 2009). We show that construction of non-malleable codes for the split-state model reduces to construction of non-malleable two-source extractors. We prove a general result on existence of seedless non-malleable extractors, which implies that codes obtained from our reduction can achieve rates arbitrarily close to 1 / 5 and exponentially small error. In a separate recent work, the authors show that the optimal rate in this model is 1 / 2. Currently, the best known explicit construction of split-state coding schemes is due to Aggarwal, Dodis and Lovett (ECCC TR13-081) which only achieves vanishing (polynomially small) rate.

Heterogeneity-Aware Codes With Uncoded Repair for Distributed Storage Systems

In practical large-scale distributed storage systems, node failures are unavoidable. It is therefore desirable to quickly recreate the failed nodes in order to maintain the system integrity. In this letter, we consider a family of erasure codes that provide uncoded repair, where the failed node is regenerated by transfer of data without extra arithmetic operations. We introduce flexible fractional repetition (FFR) code, of which the coding scheme is a concatenation of an outer MDS code and an inner repetition code. Our proposed codes are applicable to the heterogeneous network environment where node storage capacities and packet repetition degrees vary in a wide range. We present explicit constructions of FFR codes by utilizing combinatorial designs. We further propose a heuristic code construction. Evaluation results show that FFR codes outperform regenerating codes in node repair efficiency.

Insertion and deletion correcting DNA barcodes based on watermarks.

BackgroundBarcode multiplexing is a key strategy for sharing the rising capacity of next-generation sequencing devices: Synthetic DNA tags, called barcodes, are attached to natural DNA fragments within the library preparation procedure. Different libraries, can individually be labeled with barcodes for a joint sequencing procedure. A post-processing step is needed to sort the sequencing data according to their origin, utilizing these DNA labels. The final separation step is called demultiplexing and is mainly determined by the characteristics of the DNA code words used as labels.Currently, we are facing two different strategies for barcoding: One is based on the Hamming distance, the other uses the edit metric to measure distances of code words. The theory of channel coding provides well-known code constructions for Hamming metric. They provide a large number of code words with variable lengths and maximal correction capability regarding substitution errors. However, some sequencing platforms are known to have exceptional high numbers of insertion or deletion errors. Barcodes based on the edit distance can take insertion and deletion errors into account in the decoding process. Unfortunately, there is no explicit code-construction known that gives optimal codes for edit metric.ResultsIn the present work we focus on an entirely different perspective to obtain DNA barcodes. We consider a concatenated code construction, producing so-called watermark codes, which were first proposed by Davey and Mackay, to communicate via binary channels with synchronization errors. We adapt and extend the concepts of watermark codes to use them for DNA sequencing. Moreover, we provide an exemplary set of barcodes that are experimentally compatible with common next-generation sequencing platforms. Finally, a realistic simulation scenario is use to evaluate the proposed codes to show that the watermark concept is suitable for DNA sequencing applications.ConclusionOur adaption of watermark codes enables the construction of barcodes that are capable of correcting substitutions, insertion and deletion errors. The presented approach has the advantage of not needing any markers or technical sequences to recover the position of the barcode in the sequencing reads, which poses a significant restriction with other approaches.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-015-0482-7) contains supplementary material, which is available to authorized users.

Characterizing the Rate Region of the (4,3,3) Exact-Repair Regenerating Codes

Exact-repair regenerating codes are considered for the case (n,k,d)=(4,3,3), for which a complete characterization of the rate region is provided. This characterization answers in the affirmative the open question whether there exists a non-vanishing gap between the optimal bandwidth-storage tradeoff of the functional-repair regenerating codes (i.e., the cut-set bound) and that of the exact-repair regenerating codes. To obtain an explicit information theoretic converse, a computer-aided proof (CAP) approach based on primal and dual relation is developed. This CAP approach extends Yeung's linear programming (LP) method, which was previously only used on information theoretic problems with a few random variables due to the exponential growth of the number of variables in the corresponding LP problem. The symmetry in the exact-repair regenerating code problem allows an effective reduction of the number of variables, and together with several other problem-specific reductions, the LP problem is reduced to a manageable scale. For the achievability, only one non-trivial corner point of the rate region needs to be addressed in this case, for which an explicit binary code construction is given.

Every Convolutional Code is a Goppa Code

Convolutional Goppa codes (CGC) were defined in Appl. Algebra Eng. Comm. Comput., vol. 15, pp. 51-61, 2004 and IEEE Trans. Inf. Theory, vol. 52, 340-344, 2006. In this paper, we prove that every convolutional code is a CGC defined over a smooth curve over \BBF q(z) and we give an explicit construction of convolutional codes as CGC over the projective line \BBP \BBF q(z)1. We characterize which convolutional codes are defined by a complete linear system over curves of genus 0, 1, and over hyperelliptic curves. We apply these results to provide detailed constructions of some linear block codes as Goppa codes.

Coding for Secrecy: An Overview of Error-Control Coding Techniques for Physical-Layer Security

While secrecy in communication systems has historically been obtained through cryptographic means in the upper layers, recent research efforts have focused on the physical layer and have unveiled ample opportunities for security design. In particular, the combination of signal processing techniques with channel coding for secrecy has been central to the development of physical-layer security efforts. Although implicit coding techniques for secrecy have been known since the 1970s, explicit code constructions have only been discovered within the last decade. The purpose of this article is to provide a synopsis of the state of the art in coding for secrecy. We discuss the general principles of coding, and we illustrate them with several examples. In particular, we discuss the importance of a nested code structure and stochastic encoding, which allow for both data reliability and security.

Improving the DMT Performance for MIMO Communication With Linear Receivers

Multiple-input-multiple-output (MIMO) linear receivers are often of more practical interest than maximum-likelihood (ML) receivers due to their low decoding complexity but at the cost of worse diversity gain performance. Such a statement on performance loss is due to the assumption of using an independent identically distributed complex Gaussian vector as channel input. By removing this assumption, we find that the diversity performance of MIMO linear receivers can be significantly improved. In an extreme case, it can be the same as that of ML receivers. Specifically, in this paper, we investigate the diversity-multiplexing tradeoff (DMT) performance of MIMO linear receivers with colored and possibly degenerate Gaussian channel inputs. By varying the rank of the covariance matrix of the channel input vector and by allowing temporal coding across multiple channel uses, we show that the MIMO linear receiver can achieve a much better DMT performance than the currently known one. Explicit optimal code constructions are provided, along with simulation results, to justify the above findings. For the case of (2 × 2) and (3 × 3) MIMO linear receivers, simulation results show that the newly proposed codes provide significant gains of 10 and 12.08 dB in E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> at bit error rate 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> compared to the conventional schemes, respectively.

Polar Codes for Classical-Quantum Channels

Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are capacity-achieving for the task of sending classical data over a channel with classical inputs and quantum outputs. Although they demonstrated the existence of such codes, their proof does not provide an explicit construction of codes for this task. The aim of the present paper is to fill this gap by constructing near-explicit "polar" codes that are capacity-achieving. The codes exploit the channel polarization phenomenon observed by Arikan for the case of classical channels. Channel polarization is an effect in which one can synthesize a set of channels, by "channel combining" and "channel splitting," in which a fraction of the synthesized channels are perfect for data transmission while the other fraction are completely useless for data transmission, with the good fraction equal to the capacity of the channel. The channel polarization effect then leads to a simple scheme for data transmission: send the information bits through the perfect channels and "frozen" bits through the useless ones. The main technical contributions of the present paper are threefold. First, we leverage several known results from the quantum information literature to demonstrate that the channel polarization effect occurs for channels with classical inputs and quantum outputs. We then construct linear polar codes based on this effect, and the encoding complexity is O(N log N), where N is the blocklength of the code. We also demonstrate that a quantum successive cancellation decoder works well, in the sense that the word error rate decays exponentially with the blocklength of the code. For this last result, we exploit Sen's recent "non-commutative union bound" that holds for a sequence of projectors applied to a quantum state.

Secure Network Coding for Wiretap Networks of Type II

We consider the problem of securing a multicast network against a wiretapper that can eavesdrop on the packets on a limited number of network edges of its choice. We assume that the network employs network coding to simultaneously deliver the packets available at the source to all the destinations. We show that this problem can be looked at as a network generalization of the wiretap channel of type II introduced in a seminal paper by Ozarow and Wyner. In particular, we show that the transmitted information can be secured by using the Ozarow-Wyner approach of coset coding at the source on top of the existing network code. This way, we quickly and transparently recover some of the results available in the literature on secure network coding for wiretap networks. Moreover, we use this framework to derive new bounds on the code alphabet size that are independent of the network size, and provide algorithms for explicit construction of secure network codes. We also analyze the amount of information that can be leaked to the wiretapper as a function of the number of wiretapped edges.

Securing Dynamic Distributed Storage Systems Against Eavesdropping and Adversarial Attacks

We address the problem of securing distributed storage systems against eavesdropping and adversarial attacks. An important aspect of these systems is node failures over time, necessitating, thus, a repair mechanism in order to maintain a desired high system reliability. In such dynamic settings, an important security problem is to safeguard the system from an intruder who may come at different time instances during the lifetime of the storage system to observe and possibly alter the data stored on some nodes. In this scenario, we give upper bounds on the maximum amount of information that can be stored safely on the system. For an important operating regime of the distributed storage system, which we call the bandwidth-limited regime, we show that our upper bounds are tight and provide explicit code constructions. Moreover, we provide a way to short list the malicious nodes and expurgate the system.

Quasi-Cyclic LDPC Codes: Influence of Proto- and Tanner-Graph Structure on Minimum Hamming Distance Upper Bounds

Quasi-cyclic (QC) low-density parity-check (LDPC) codes are an important instance of proto-graph-based LDPC codes. In this paper we present upper bounds on the minimum Hamming distance of QC LDPC codes and study how these upper bounds depend on graph structure parameters (like variable degrees, check node degrees, girth) of the Tanner graph and of the underlying proto-graph. Moreover, for several classes of proto-graphs we present explicit QC LDPC code constructions that achieve (or come close to) the respective minimum Hamming distance upper bounds. Because of the tight algebraic connection between QC codes and convolutional codes, we can state similar results for the free Hamming distance of convolutional codes. In fact, some QC code statements are established by first proving the corresponding convolutional code statements and then using a result by Tanner that says that the minimum Hamming distance of a QC code is upper bounded by the free Hamming distance of the convolutional code that is obtained by "unwrapping" the QC code.

DMT Optimal Codes Constructions for Multiple-Access MIMO Channel

Explicit code constructions for multiple-input multiple-output (MIMO) multiple-access channels (MAC) with K users are presented in this paper. The first construction is dedicated to the case of symmetric MIMO-MAC where all the users have the same number of transmit antennas nt and transmit at the same level of per-user multiplexing gain r. Furthermore, we assume that the users transmit in an independent fashion and do not cooperate. The construction is systematic for any values of K, nt and r. It is proved that this newly proposed construction achieves the optimal MIMO-MAC diversity-multiplexing gain tradeoff (DMT) provided by Tse at high-SNR regime. In the second part of the paper we take a further step to investigate the MAC-DMT of a general MIMO-MAC where the users are allowed to have different numbers of transmit antennas and can transmit at different levels of multiplexing gain. The exact optimal MAC-DMT of such channel is explicitly characterized in this paper. Interestingly, in the general MAC-DMT, some users might not be able to achieve their single-user DMT performance as in the symmetric case, even when the multiplexing gains of the other users are close to 0. Detailed explanations of such unexpected result are provided in this paper. Finally, by generalizing the code construction for the symmetric MIMO-MAC, explicit code constructions are provided for the general MIMO-MAC and are proved to be optimal in terms of the general MAC-DMT.

Full-Diversity Space-Time Error Correcting Codes with Low-Complexity Receivers

We propose an explicit construction of full-diversity space-time block codes, under the constraint of an error correction capability. Furthermore, these codes are constructed in order to be suitable for a serial concatenation with an outer linear forward error correcting (FEC) code. We apply the binary rank criterion, and we use the threaded layering technique and an inner linear FEC code to define a space-time error-correcting code. When serially concatenated with an outer linear FEC code, a product code can be built at the receiver, and adapted iterative receiver structures can be applied. An optimized hybrid structure mixing MMSE turbo equalization and turbo product code decoding is proposed. It yields reduced complexity and enhanced performance compared to previous existing structures.