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Biometric authentication systems, based on secret key generation, work as follows. In the enrollment stage, an individual provides a biometric signal that is mapped into a secret key and a helper message, the former being prepared to become available to the system at a later time (for authentication), and the latter is stored in a public database. When an authorized user requests authentication, claiming his/her identity to be one of those of the subscribers, he/she has to provide a biometric signal again, and then the system, which retrieves also the helper message of the claimed subscriber, produces an estimate of the secret key that is finally compared to the secret key of the claimed user. In the case of a match, the authentication request is approved; otherwise, it is rejected. Evidently, there is an inherent tension between two desired, but conflicting, properties of the helper message encoder; on the one hand, the encoding should be informative enough concerning the identity of the real subscriber, in order to approve him/her in the authentication stage, but on the other hand, it should not be too informative, as otherwise, unauthorized imposters could easily fool the system and gain access. A good encoder should then trade off the two kinds of errors: the false reject (FR) error and the false accept (FA) error. In this paper, we investigate trade-offs between the random coding FR error exponent and the best achievable FA error exponent. We compare two types of ensembles of codes: fixed-rate codes and variable-rate codes, and we show that the latter class of codes offers considerable improvement compared to the former. In doing this, we characterize ensemble-optimal rate functions for both types of codes. We also examine the effect of privacy leakage constraints for both fixed-rate codes and variable-rate codes [ABSTRACT FROM AUTHOR]

Leafless elementary trapping sets (LETSs) are known to be the problematic structures in the error floor region of low-density parity-check (LDPC) codes over the additive white Gaussian (AWGN) channel under iterative decoding algorithms. While problems involving the general category of trapping sets, and the subcategory of elementary trapping sets (ETSs), have been shown to be NP-hard, similar results for LETSs, which are a subset of ETSs are not available. In this paper, we prove that for a general LDPC code, finding a LETS of a given size $a$ with minimum number of odd-degree check nodes $b$ is NP-hard to approximate within any approximation factor. We also prove that finding the minimum size $a$ of a LETS with a given $b$ is NP-hard to approximate within any approximation factor. Similar results are proved for elementary absorbing sets, a popular subcategory of LETSs. [ABSTRACT FROM AUTHOR]

The input-constrained binary erasure channel (BEC) with strictly causal feedback is studied. The channel input sequence must satisfy the $(0,k)$ -runlength limited (RLL) constraint, i.e., no more than $k$ consecutive ‘0’s are allowed. The feedback capacity of this channel is derived for all $k\geq 1$ , and is given by $C^{\mathrm {fb}}_{(0,k)}(\varepsilon) = \max \frac {\overline {\varepsilon }H_{2}(\delta _{0})+\sum _{i=1}^{k-1}\left ({\overline {\varepsilon }^{i+1}H_{2}(\delta _{i})\prod _{m=0}^{i-1}\delta _{m}}\right)}{1+\sum _{i=0}^{k-1}\left ({\overline {\varepsilon }^{i+1} \prod _{m=0}^{i}\delta _{m}}\right)}$ , where $\varepsilon $ is the erasure probability, $\overline {\varepsilon }=1-\varepsilon $ and $H_{2}(\cdot)$ is the binary entropy function. The maximization is only over $\delta _{k-1}$ , while the parameters $\delta _{i}$ for $i\leq k-2$ are straightforward functions of $\delta _{k-1}$. The lower bound is obtained by constructing a simple coding for all $k\geq 1$. It is shown that the feedback capacity can be achieved using zero-error, variable length coding. For the converse, an upper bound on the non-causal setting, where the erasure is available to the encoder just prior to the transmission, is derived. This upper bound coincides with the lower bound and concludes the search for both the feedback capacity and the non-causal capacity. As a result, non-causal knowledge of the erasures at the encoder does not increase the feedback capacity for the $(0,k)$ -RLL input-constrained BEC. This property does not hold in general: the $(2,\infty)$ -RLL input-constrained BEC, where every ‘1’ is followed by at least two ‘0’s, is used to show that the feedback capacity can be strictly smaller than the non-causal capacity. [ABSTRACT FROM AUTHOR]

This paper considers transmitting a sequence of messages (streaming messages) over a packet erasure channel. In each time slot, the source constructs a packet based on the current and the previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. Every source message must be recovered perfectly at the destination subject to a fixed decoding delay. We assume that the channel loss model introduces either one burst erasure or multiple arbitrary erasures in any fixed-sized sliding window. Under this channel loss assumption, we fully characterize the maximum achievable rate by constructing streaming codes that achieve the optimal rate. In addition, our construction of optimal streaming codes implies the full characterization of the maximum achievable rate for convolutional codes with any given column distance, column span, and decoding delay. Numerical results demonstrate that the optimal streaming codes outperform existing streaming codes of comparable complexity over some instances of the Gilbert–Elliott channel and the Fritchman channel. [ABSTRACT FROM AUTHOR]

In this paper, we consider a binary erasure channel (BEC) with an unknown erasure probability of ${\delta }$ at a transmitter. In addition, we consider that ${\delta }$ has a constant value in each transmission block and the transmitter knows the probability distribution of ${\delta }$. For this problem with the infinite block length, based on the distribution of the random variable ${\delta }$ , a multi-layer encoding strategy at the transmitter and a successive decoding strategy at the receiver are proposed. In our proposed scheme, based on the value of ${\delta }$ , the receiver can decode a part of transmitted data from different transmission layers. In order to have a comparison with other erasure correction codes, we generalize the result of the infinite block length to the finite block-length transmission. We show that considering the average transmission rate, our scheme has a better performance than the other erasure correction codes such as the Raptor and the MDS codes. We also analyze both the decoding complexity and bit error rate versus the erasure probability of the channel. Besides, by using a numerical comparison, we try to show that our proposed strategy outperforms other transmission strategies. [ABSTRACT FROM AUTHOR]

Sampling from the lattice Gaussian distribution plays an important role in various research fields. In this paper, the Markov chain Monte Carlo (MCMC)-based sampling technique is advanced in several fronts. First, the spectral gap for the independent Metropolis-Hastings-Klein (MHK) algorithm is derived, which is then extended to Peikert’s algorithm and rejection sampling; we show that independent MHK exhibits faster convergence. Then, the performance of bounded distance decoding (BDD) using MCMC is analyzed, revealing a flexible trade-off between the decoding radius and complexity. MCMC is further applied to trapdoor sampling, again offering a trade-off between security and complexity. Finally, the independent multiple-try Metropolis-Klein (MTMK) algorithm is proposed to enhance the convergence rate. The proposed algorithms allow parallel implementation, which is beneficial for practical applications. [ABSTRACT FROM AUTHOR]

Protograph-based Raptor-like low-density parity-check codes (PBRL codes) are a family of easily encodable rate-compatible low-density parity-check (LDPC) codes. PBRL codes have an excellent performance across all design rates. Quasi-cyclic (QC) PBRL code families permit high-speed decoder implementations. PBRL codes designed thus far, for both long and short block-lengths, have been based on optimizing the iterative decoding threshold of the protograph of the PBRL family at various design rates. This paper introduces a design method to obtain better QC PBRL code families at short block-lengths (of a few hundred bits) for low frame error rate (FER) requirements. We first select a protomatrix for the highest design rate. To add a new row to lower the rate, we keep all the previously obtained rows of the PBRL protomatrix fixed and select the new row that maximizes an upper bound on the minimum distance of any QC-LDPC code that can be obtained from the protomatrix. The new QC PBRL code families outperform the original PBRL codes at short block-lengths by providing a significantly better low-FER performance. The standard approach to computing the aforementioned upper bounds requires complexity that grows exponentially with the size of the protomatrix. However, we show that the structure of the PBRL protomatrix lets us obtain the upper bounds with complexity that grows only linearly with the size of the PBRL protomatrix. Using the complexity reduction results, we also establish an equivalence between the exhaustive search to design a new row for the PBRL protomatrix according to the new design method and an integer linear program. [ABSTRACT FROM AUTHOR]

Partial syndrome matrices and weak-locator polynomials have received considerable attention in recent years due to their applications in decoding cyclic codes. The technical contribution of this paper is threefold: 1) cyclic codes can be decoded by the newly proposed partial syndrome matrices, which generalize the previously known results on the determination of error positions; 2) a new type of polynomial associated with error locations, called radical-locator polynomial, is defined, which includes the weak-locator polynomial as a special case; and 3) a novel class of matrices with many zero entries, called row-echelon partial syndrome matrices, is presented, based on the Newton identities, for efficiently decoding cyclic codes. It is also shown that the radical-locator polynomials can be obtained from the determinants of the above-mentioned (row-echelon) partial syndrome matrices. [ABSTRACT FROM AUTHOR]

The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of possibly noisy tests, and is relevant in applications such as medical testing, communication protocols, pattern matching, and many more. One of the defining features of the group testing problem is the distinction between the non-adaptive and adaptive settings. In the non-adaptive case, all tests must be designed in advance, whereas in the adaptive case, each test can be designed based on the previous outcomes. While tight information-theoretic limits and near-optimal practical algorithms are known for the adaptive setting in the absence of noise, surprisingly little is known in the noisy adaptive setting. In this paper, we address this gap by providing information-theoretic achievability and converse bounds under various noise models, as well as a slightly weaker achievability bound for a computationally efficient variant. These bounds are shown to be tight or near-tight in a broad range of scaling regimes, particularly at low noise levels. The algorithms used for the achievability results have the notable feature of only using two or three stages of adaptivity. [ABSTRACT FROM AUTHOR]

Mutually uncorrelated (MU) codes are a class of codes in which no proper prefix of one codeword is a suffix of another codeword. These codes were originally studied for synchronization purposes and recently, Yazdi et al. showed their applicability to enable random access in DNA storage. In this paper,we follow the research of Yazdi et al. and study MU codes along with their extensions to correct errors and balanced codes. We first review a well-known construction of MU codes and study the asymptotic behavior of its cardinality. This task is accomplished by studying a special class of run-length limited codes that impose the longest run of zeros to be at most some function of the codewords length. We also present an efficient algorithm for this class of constrained codes and show how to use this analysis for MU codes. Next, we extend the results on the run-length limited codes in order to study $(d_{h},d_{m})$ -MU codes that impose a minimum Hamming distance of $d_{h}$ between different codewords and $d_{m}$ between prefixes and suffixes. In particular, we show an efficient construction of these codes with nearly optimal redundancy. We also provide similar results for the edit distance and balanced MU codes. Last, we draw connections to the problems of comma-free and prefix synchronized codes. [ABSTRACT FROM AUTHOR]

It is a well-known fact that feedback does not increase the capacity of point-to-point memoryless channels, however, its effect in secure communications is not fully understood yet. In this paper, an achievable scheme for the wiretap channel with generalized feedback is presented. This scheme, which uses the feedback signal to generate a shared secret key between the legitimate users, encrypts the message to be sent at the bit level. New capacity results for a class of channels are provided, as well as some new insights into the secret key agreement problem. Moreover, this scheme recovers previously reported rate regions from the literature, and thus it can be seen as a generalization that unifies several results in the field. [ABSTRACT FROM AUTHOR]

We consider data transmission over a network where each edge is an erasure channel and the inner nodes transmit the random linear combinations of their incoming information. We distinguish two channel models in this setting: the row and the column erasure channel model. For both models, we investigate spread codes and determine their symbol erasure correction capability and the probability of decoding success. We also compare the performance of spread codes to other known codes suitable for those models. Furthermore, we explain how to decode these codes in the two channel models and compare the decoding complexities. The results show that depending on the application and the to-be-optimized aspect, any combination of codes and channel models can be the best choice. [ABSTRACT FROM AUTHOR]

Existing polar coding schemes for the two-user interference channel follow the original idea of Han and Kobayashi, in which component messages are encoded independently and then mapped by some deterministic functions (i.e., homogeneous superposition coding). In this paper, we propose a new polar coding scheme for the interference channel based on the heterogeneous superposition coding approach of Chong, Motani, and Garg. We prove that fully joint decoding (the receivers simultaneously decode both senders’ common messages and the intended sender’s private message) in the Han–Kobayashi strategy can be simplified to two types of partial-joint decoding, which are friendly to polar coding with practical decoding algorithms. The proposed coding scheme requires less auxiliary random variables and no deterministic functions and can be efficiently constructed. Furthermore, we extend this result to interference networks and show that partial-joint decoding is a general method for designing heterogeneous superposition polar coding schemes in interference networks. [ABSTRACT FROM AUTHOR]

Gaussian thermal loss channels are of particular importance to quantum communication theory since they model realistic optical communication channels. Except for special cases, the quantum capacity of Gaussian thermal loss channels is not yet quantified completely. In this paper, we provide improved upper bounds of the Gaussian thermal loss channel capacity, both in energy-constrained and unconstrained scenarios. We briefly review Gottesman-Kitaev-Preskill (GKP) codes and discuss their experimental implementation. We then prove, in the energy-unconstrained case, that a family of GKP codes achieves the quantum capacity of Gaussian thermal loss channels up to at most a constant gap from the improved upper bound. In the energy-constrained case, we formulate a biconvex encoding and decoding optimization problem to maximize entanglement fidelity. Then, we solve the biconvex optimization heuristically by an alternating semi-definite programming method and report that, starting from Haar random initial codes, our numerical optimization yields a hexagonal GKP code as an optimal encoding in a practically relevant regime. [ABSTRACT FROM AUTHOR]

Every channel can be expressed as a convex combination of deterministic channels with each deterministic channel corresponding to one particular intrinsic state. Such convex combinations are, in general, not unique, each giving rise to a specific intrinsic-state distribution. In this paper, we study the maximum and minimum capacities of a channel when the realization of its intrinsic state is causally available at the encoder and/or the decoder. Several conclusive results are obtained for binary-input channels and binary-output channels. By-products of our investigation include a generalization of the Birkhoff–von Neumann theorem and a condition on the uselessness of causal state information at the encoder. [ABSTRACT FROM AUTHOR]

Algorithms for secure encryption in a post-quantum world are currently receiving a lot of attention in the research community. One of the most promising such algorithms is the code-based scheme called QC-MDPC, which has excellent performance and a small public key size. In this paper, we present a very efficient key recovery attack on the QC-MDPC scheme using the fact that decryption uses an iterative decoding step, and this can fail with some small probability. We identify a dependence between the secret key and the failure in decoding. This can be used to build what we refer to as a distance spectrum for the secret key, which is the set of all distances between any two ones in the secret key. In a reconstruction step, we then determine the secret key from the distance spectrum. The attack has been implemented and tested on a proposed instance of QC-MDPC for 80-bit security. It successfully recovers the secret key in minutes. A slightly modified version of the attack can be applied on proposed versions of the QC-MDPC scheme that provides IND-CCA security. The attack is a bit more complex in this case, but still very much below the security level. The reason why we can break schemes with proved CCA security is that the model for these proofs typically does not include the decoding error possibility. At last, we present several algorithms for key reconstruction from an empirical distance spectrum. We first improve the naïve algorithm for key reconstruction by a factor of about 3 0000, when the parameters for 80-bit security are implemented. We further develop the algorithm to deal with errors in the distance spectrum. This ultimately reduces the requirement on the number of ciphertexts that need to be collected for a successful key recovery. [ABSTRACT FROM AUTHOR]

In this paper, we show that any sequence of infinite lattice constellations which is good for the unconstrained Gaussian channel can be shaped into a capacity-achieving sequence of codes for the power-constrained Gaussian channel under lattice decoding and non-uniform signaling. Unlike previous results in the literature, our scheme holds with no extra condition on the lattices (e.g., quantization-goodness or vanishing flatness factor), thus establishing a direct implication between AWGN-goodness, in the sense of Poltyrev and capacity-achieving codes. Our analysis uses properties of the discrete Gaussian distribution in order to obtain precise bounds on the probability of error and achievable rates. In particular, we obtain a simple characterization of the finite-blocklength behavior of the scheme, showing that it approaches the optimal dispersion coefficient for high signal-to-noise ratio. We further show that for low signal-to-noise ratio, the discrete Gaussian over centered lattice constellations cannot achieve capacity, and thus a shift (or “dither”) is essentially necessary. [ABSTRACT FROM AUTHOR]

The Korkine–Zolotareff (KZ) reduction is one of the often used reduction strategies for lattice decoding. In this paper, we first investigate some important properties of KZ reduced matrices. Specifically, we present a linear upper bound on the Hermit constant which is around 7/8 times of the existing sharpest linear upper bound, and an upper bound on the KZ constant which is polynomially smaller than the existing sharpest one. We also propose upper bounds on the lengths of the columns of KZ reduced matrices, and an upper bound on the orthogonality defect of KZ reduced matrices which are even polynomially and exponentially smaller than those of boosted KZ reduced matrices, respectively. Then, we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem (SVP) when its basis matrix is LLL reduced. These upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a KZ reduction algorithm. Finally, we propose a new KZ reduction algorithm by modifying the commonly used Schnorr–Euchner search strategy for solving SVPs and the basis expansion method proposed by Zhang et al. Simulation results show that the new KZ reduction algorithm is much faster and more numerically reliable than the KZ reduction algorithm proposed by Zhang et al., especially when the basis matrix is ill conditioned. [ABSTRACT FROM AUTHOR]

Communication in integrated circuits faces two major impediments: inter-wire capacitive coupling and noise. Coding can be used to address both these problems. So-called crosstalk-avoidance codes mitigate capacitive coupling, and traditional error-correction codes introduce resilience against channel errors. Unfortunately, crosstalk-avoidance and error-correction codes cannot be combined in a straightforward manner. On the one hand, crosstalk-avoidance encoding followed by error-correction encoding destroys the crosstalk-avoidance property. On the other hand, error-correction encoding followed by crosstalk-avoidance encoding causes the crosstalk-avoidance decoder to fail in the presence of errors. Existing approaches circumvent this difficulty by using additional bus wires to protect the parities generated from the output of the error-correction encoder, and are therefore inefficient. In this paper, we propose a novel joint crosstalk-avoidance and error-correction coding and decoding scheme that provides higher bus transmission rates compared with existing approaches. Our joint approach carefully embeds the parities such that the crosstalk-avoidance property is preserved. We analyze the rate and minimum distance of the proposed scheme. We also provide a density evolution analysis and predict iterative decoding thresholds for reliable communication under random bus erasures. This density evolution analysis is nonstandard, since the crosstalk-avoidance constraints are inherently nonlinear. [ABSTRACT FROM AUTHOR]

In this paper, we propose a characterization for non-elementary trapping sets (NETSs) of low-density parity-check (LDPC) codes. The characterization is based on viewing an NETS as a hierarchy of embedded graphs starting from an ETS. The characterization corresponds to an efficient search algorithm that under certain conditions is exhaustive. As an application of the proposed characterization/search, we obtain lower and upper bounds on the stopping distance $s_{\min }$ of LDPC codes. We examine a large number of regular and irregular LDPC codes and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, $s_{\min }$. Finding $s_{\min }$ , or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm. For a constant degree distribution and range of search, the worst case computational complexity of the proposed search algorithms for finding NETSs and stopping sets is linear in the code’s block length $n$. The average search complexity for stopping sets, however, is constant in $n$ , if the simple cycles required as input to the search algorithm are available. [ABSTRACT FROM AUTHOR]

We propose two classes of universal codes that are suited to two asymptotic regimes when the output alphabet is possibly continuous. The first class has the property that the error probability decays exponentially fast, and we identify an explicit lower bound on the error exponent. The other class attains the epsilon-capacity of the channel, and we also identify the second-order term in the asymptotic expansion. The proposed encoder is essentially based on the packing lemma of the method of types. For the decoder, we first derive a Rényi-relative-entropy version of Clarke and Barron’s formula the distance between the true distribution and the Bayesian mixture, which is of independent interest. The universal decoder is stated in terms of this formula and quantities used in the information spectrum method. The methods contained herein allow us to analyze universal codes for channels with continuous and discrete output alphabets in a unified manner and to analyze their performances in terms of the exponential decay of the error probability and the second-order coding rate. [ABSTRACT FROM AUTHOR]

We study a multihop “virtual” full-duplex relay channel as a special case of a general multiple multicast relay network. For such a channel, quantize-map-and-forward (QMF) [and its generalization of noisy network coding (NNC) and short message NNC] achieves the cut-set upper bound within a constant additive gap, where the gap grows linearly with the number of relay stages $K$. This gap, however, may not be acceptable for practical communication systems with multihop transmissions (e.g., a wireless backhaul operating at high frequencies). Recently, we improved the capacity scaling by using a forward sliding-window (SW) decoding and by optimizing the quantization level at each relay, obtaining the gap that grows logarithmically as $\log {K}$. Furthermore, the improved scheme has lower decoding complexity and delay than the general QMF and NNC approaches. In this paper, we further improve the performance by presenting a mixed scheme in which each relay can perform either decode-and-forward (DF) or the improved QMF (with SW decoding) and can choose to perform rate-splitting to enable partial interference cancellation. In general, the optimization of the relay DF/QMF configuration is combinatorial. Nevertheless, we provide that a simple greedy algorithm finds an optimal configuration under some practically reasonable assumptions. We derive an achievable rate that is easily computable and show that the proposed mixed scheme outperforms the QMF-only schemes. We demonstrate that the performance improvement increases with $K$ , which indicates that the mixed scheme is indeed beneficial for multihop transmission. [ABSTRACT FROM AUTHOR]

An error-erasure channel is a simple noise model that introduces both errors and erasures. While the two types of errors can be corrected simultaneously with error-correcting codes, it is also known that any linear code allows for first correcting errors and then erasures in two-step decoding. In particular, a carefully designed parity-check matrix not only allows for separating erasures from errors but also makes it possible to efficiently correct erasures. The separating redundancy of a linear code is the number of parity-check equations in a smallest parity-check matrix that has the required property for this error-erasure separation. In a sense, it is a parameter of a linear code that represents the minimum overhead for efficiently separating erasures from errors. While several bounds on separating redundancy are known, there still remains a wide gap between upper and lower bounds except for a few limited cases. In this paper, using probabilistic combinatorics and design theory, we improve both upper and lower bounds on separating redundancy. We also show a relation between parity-check matrices for error-erasure separation and special matrices, called X-codes, for data compaction circuits in VLSI testing. This leads to an exponentially improved bound on the size of an optimal X-code. [ABSTRACT FROM AUTHOR]

We investigate the maximum coding rate for a given average blocklength and error probability over a $K$ -user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. For the point-to-point case, Polyanskiyet al.(2011) demonstrated that variable-length coding combined with stop-feedback significantly increases the speed of convergence of the maximum coding rate to capacity. This speed-up manifests itself in the absence of a square-root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths, i.e.,zero dispersion. In this paper, we present nonasymptotic achievability and converse bounds on the maximum coding rate of the common-message $K$ -user discrete memoryless broadcast channel, which strengthen and generalize the ones reported in Trillingsgaardet al.(2015) for the two-user case. An asymptotic analysis of these bounds reveals that zero dispersion cannot be achieved for certain common-message broadcast channels (e.g., the binary symmetric broadcast channel). Furthermore, we identify conditions under which our converse and achievability bounds are tight up to the second order. Through numerical evaluations, we illustrate that our second-order expansions approximate accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point-to-point case. [ABSTRACT FROM AUTHOR]

The polarization process of polar codes over a prime $q$ -ary alphabet is studied. Recently, it has been shown that the blocklength of polar codes with prime alphabet size scales polynomially with respect to the inverse of the gap between code rate and channel capacity. However, except for the binary case, the degree of the polynomial in the bound is extremely large. In this paper, a different approach to computing the degree of this polynomial for any prime alphabet size is shown. This approach yields a lower degree polynomial for various values of the alphabet size that were examined. It is also shown that even lower degree polynomial can be computed with an additional numerical effort. [ABSTRACT FROM AUTHOR]

We consider the dynamics of message passing for spatially coupled codes and, in particular, the set of density evolution equations that tracks the profile of decoding errors along the spatial direction of coupling. It is known that, for suitable boundary conditions and after a transient phase, the error profile exhibits a “solitonic behavior.” Namely, a uniquely shaped wavelike solution develops, which propagates with a constant velocity. Under this assumption, we derive an analytical formula for the velocity in the framework of a continuum limit of the spatially coupled system. The general formalism is developed for spatially coupled low-density parity-check codes on general binary memoryless symmetric channels, which form the main systems of interest in this paper. We apply the formula for special channels and illustrate that it matches the direct numerical evaluation of the velocity for a wide range of noise values. A possible application of the velocity formula to the evaluation of finite size scaling law parameters is also discussed. We conduct a similar analysis for general scalar systems and illustrate the findings with applications to compressive sensing and generalized low-density parity-check codes on the binary erasure or binary symmetric channels. [ABSTRACT FROM AUTHOR]

In this paper, we prove a unified achievability bound that generalizes and improves random coding bounds for any combination of source coding, channel coding, joint source–channel coding, and coding for computing problems assuming blockwise node operation. As a general network setup, we consider an acyclic discrete memoryless network, where the network demands and constraints are specified by a joint-typicality constraint on the whole channel input and output sequences. For achievability, a basic building block for node operation consists of simultaneous nonunique decoding, simultaneous compression, and symbol-by-symbol mapping. Our bound can be useful for deriving random coding bounds without error analysis, especially for large and complex networks. In particular, our bound can be used for unifying and generalizing many known relaying strategies. For example, a generalized decode-compress-amplify-and-forward bound is obtained as a simple corollary of our main theorem, and it is shown to strictly outperform the previously known relaying schemes. Furthermore, by exploiting the symmetry in our bound, we formally define and characterize three types of network duality based on channel input–output reversal and network flow reversal combined with packing–covering duality. [ABSTRACT FROM AUTHOR]

The ordering of communication channels was first introduced by Shannon. In this paper, we aim to find characterizations of two orderings: input-degradedness and the Shannon ordering. A channel $W$ is said to be input-degraded from another channel $W'$ if $W$ can be simulated from $W'$ by randomization at the input. We provide a necessary and sufficient condition for a channel to be input-degraded from another one. We show that any decoder that is good for $W$ is also good for $W'$. We provide two characterizations for input-degradedness, one of which is similar to the Blackwell-Sherman-Stein theorem. We show that $W'$ contains $W$ (in the Shannon ordering sense) if and only if $W$ is the skew-composition of $W'$ with a convex-product $n$ channel. This fact is used to derive a characterization of the Shannon ordering that is similar to the Blackwell-Sherman-Stein theorem. [ABSTRACT FROM AUTHOR]

A class of cyclic codes has recently been decoded by using the weak-locator polynomials instead of the conventional error-locator polynomials. In this paper, a generalization of weak-locator polynomials to the multivariate cases, called the multivariate weak-locator polynomials, is defined, and a new matrix whose determinant can be expressed as a multivariate weak-locator polynomial is developed for cyclic codes. Moreover, a modified matrix along with Gaussian elimination found herein enables one to determine error positions precisely. The presented matrices result in a reduction of the number of syndromes when compared with the previously known matrices. The simulation shows that, for example, with the capability of correcting more errors, a considerably sophistical scheme of decoding the (97, 49, 15) binary cyclic code based on the proposed matrices is around 115 times faster than other existing decoders. Therefore, in general, it is developed to facilitate faster decoding of a large class of cyclic codes and is naturally suitable for the software implementation. [ABSTRACT FROM AUTHOR]

Compressed sensing is now established as an effective method for dimension reduction when the underlying signals are sparse or compressible with respect to some suitable basis or frame. One important, yet under-addressed problem regarding the compressive acquisition of analog signals is how to perform quantization. This is directly related to the important issues of how “compressed” compressed sensing is (in terms of the total number of bits one ends up using after acquiring the signal) and ultimately whether compressed sensing can be used to obtain compressed representations of suitable signals. In this paper, we propose a concrete and practicable method for performing “analog-to-information conversion”. Following a compressive signal acquisition stage, the proposed method consists of a quantization stage, based on $ \Sigma \Delta $ (sigma-delta) quantization, and a subsequent encoding (compression) stage that fits within the framework of compressed sensing seamlessly. We prove that, using this method, we can convert analog compressive samples to compressed digital bitstreams and decode using tractable algorithms based on convex optimization. We prove that the proposed analog-to-information converter (AIC) provides a nearly optimal encoding of sparse and compressible signals. Finally, we present numerical experiments illustrating the effectiveness of the proposed AIC. [ABSTRACT FROM AUTHOR]

This paper proposes the Guruswami-Sudan (GS) list decoding algorithm for one-point elliptic codes, in which the interpolation is realized by the module basis reduction (BR). Elliptic codes are a kind of algebraic-geometric (AG) codes with a genus of one. Over the same finite field, they have a greater codeword length than Reed-Solomon (RS) codes, capable of correcting more errors. The GS decoding consists of interpolation and root-finding, while the former that determines the interpolation polynomial $\mathcal {Q}(\text {x}, \text {y}, \text {z})$ dominates the decoding complexity. By defining the Lagrange interpolation function over an elliptic function field, a basis of the interpolation module can be constructed. The desired Gröbner basis that contains $\mathcal {Q}(\text {x}, \text {y}, \text {z})$ can be determined by reducing the constructed basis. This is namely the BR interpolation and it requires less finite field arithmetic operations than the conventional Kötter’s interpolation, facilitating the GS decoding. Re-encoding transform (ReT) is further introduced to facilitate the BR interpolation. This work also shows that both the BR interpolation and its ReT variant will have a lower complexity as the code rate ${k}/{n}$ increases, where n and ${k}$ are the length and dimension of the code, respectively. Our numerical results demonstrate the complexity advantage of the BR interpolation over Kötter’s interpolation, and the performance advantage of elliptic codes over RS codes. [ABSTRACT FROM AUTHOR]

The basic goal of threshold group testing is to identify up to $d$ defective items among a population of $n$ items, where $d$ is usually much smaller than $n$. The outcome of a test on a subset of items is positive if the subset has at least $u$ defective items, negative if it has up to $\ell $ defective items, where $0 \leq \ell < u$ , and arbitrary otherwise. This is called threshold group testing. The parameter $g = u - \ell - 1$ is called the gap. In this paper, we focus on the case $g > 0$ , i.e., threshold group testing with a gap. Note that the results presented here are also applicable to the case $g = 0$ ; however, the results are not as efficient as those in related work. Currently, a few reported studies have investigated test designs and decoding algorithms for identifying defective items. Most of the previous studies have not been feasible because there are numerous constraints on their problem settings or the decoding complexities of their proposed schemes are relatively large. Therefore, it is compulsory to reduce the number of tests as well as the decoding complexity, i.e., the time for identifying the defective items, for achieving practical schemes. The work presented here makes five contributions. The first is a more accurate theorem for a non-adaptive algorithm for threshold group testing proposed by Chen and Fu. The second is an improvement in the construction of disjunct matrices, which are the main tools for tackling (threshold) group testing and other tasks such as constructing cover-free families or learning hidden graphs. Specifically, we present a better exact upper bound on the number of tests for disjunct matrices compared with that in related work. The third and fourth contributions are a reduced exact upper bound on the number of tests and a reduced asymptotic bound on the decoding time for identifying defective items in a noisy setting on test outcomes. The fifth contribution is a simulation on the number of tests of the resulting improvements for previous work and the proposed theorems. [ABSTRACT FROM AUTHOR]

A Goppa code over $\Bbb F_{q^{m}}$ is a well-known subclass of algebraic error-correcting code. If $m=1$ , then it is a generalized Reed-Solomon(GRS) code and its dual code is called a GRS code via a Goppa code. In this paper, we give a necessary and sufficient condition that the dual codes of GRS codes via (expurgated) Goppa codes are also GRS codes via Goppa codes. Under the above condition, we show that the hulls of GRS codes via Goppa codes are still GRS codes via Goppa codes. As an application, we characterize LCD GRS codes and self-dual GRS codes under the above condition. Some numerical examples are also presented to illustrate our main results. Moreover, we also apply our result to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs with arbitrary parameters. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the error floor of quasi-cyclic (QC) low-density parity-check (LDPC) codes decoded by the sum-product algorithm (SPA) with row layered message-passing scheduling. For this, we develop a linear state-space model of trapping sets (TSs) which incorporates the layered nature of scheduling. We demonstrate that the contribution of each TS to the error floor is not only a function of the topology of the TS, but also depends on the row layers in which different check nodes of the TS are located. This information, referred to as TS layer profile (TSLP), plays an important role in the harmfulness of a TS. As a result, the harmfulness of a TS in particular, and the error floor of the code in general, can significantly change by changing the order in which the information of different layers, corresponding to different row blocks of the parity-check matrix, is updated. We also study the problem of finding a layer ordering that minimizes the error floor, and obtain row layered decoders with error floor significantly lower than that of their flooding counterparts. As part of our analysis, we make connections between the parameters of the state-space model for a row layered schedule and those of the flooding schedule. Simulation results are presented to show the accuracy of analytical error floor estimates. [ABSTRACT FROM AUTHOR]

In this paper, we consider the list decoding property of codes under insertion and deletion errors (insdel for short). Firstly, we analyse the list decodability of random insdel codes. Our result provides a more complete picture on the list decodability of insdel codes when both insertion and deletion errors happen. Secondly, we construct a family of insdel codes along with their efficient encoding and decoding algorithms through concatenation method which provides a Zyablov-type bound for insdel metric codes. [ABSTRACT FROM AUTHOR]

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]

Consider a (multiple-access) wireless communication system where users are connected to a unique base station over a shared-spectrum radio links. Each user has a fixed number k of bits to send to the base station, and his signal gets attenuated by a random channel gain (quasi-static fading). In this paper we consider the many-user asymptotics of Chen-Chen-Guo’2017, where the number of users grows linearly with the blocklength. Differently, though, we adopt a per-user probability of error (PUPE) criterion (as opposed to classical joint-error probability criterion). Under PUPE the finite energy-per-bit communication is possible, and we are able to derive bounds on the tradeoff between energy and spectral efficiencies. We reconfirm the curious behaviour (previously observed for non-fading MAC) of the possibility of almost perfect multi-user interference (MUI) cancellation for user densities below a critical threshold. Further, we demonstrate the suboptimality of standard solutions such as orthogonalization (i.e. TDMA/FDMA) and treating interference as noise (i.e. pseudo-random CDMA without multi-user detection). Notably, the problem treated here can be seen as a variant of support recovery in compressed sensing for the unusual definition of sparsity with one non-zero entry per each contiguous section of 2k coordinates. This identifies our problem with that of the sparse regression codes (SPARCs) and hence our results can be equivalently understood in the context of SPARCs with sections of length 2100. Finally, we discuss the relation of the almost perfect MUI cancellation property and the replica-method predictions. [ABSTRACT FROM AUTHOR]

Consider an energy-harvesting receiver that uses the same received signal both for decoding information and for harvesting energy, which is employed to power its circuitry. In the scenario where the receiver has limited battery size, a signal with bursty energy content may cause power outage at the receiver, since the battery will drain during intervals with low signal energy. In this paper, we analyze subblock energy-constrained codes (SECCs), which ensure that sufficient energy is carried within every subblock duration. We consider discrete memoryless channels and characterize the SECC capacity and the SECC error exponent, and provide useful bounds for these values. We also study constant subblock-composition codes (CSCCs), which are a subclass of SECCs where all the subblocks in every codeword have the same fixed composition, and this subblock composition is chosen to maximize the rate of information transfer while meeting the energy requirement. Compared with constant composition codes (CCCs), we show that CSCCs incur a rate loss and that the error exponent for CSCCs is also related to the error exponent for CCCs by the same rate loss term. We exploit the symmetry in CSCCs to obtain a necessary and sufficient condition on the subblock length for avoiding power outage at the receiver. Furthermore, for CSCC sequences, we present a tight lower bound on the average energy per symbol within a sliding time window. We provide numerical examples highlighting the tradeoff between the delivery of sufficient energy to the receiver and achieving high information transfer rates. It is observed that the ability to use energy in real-time imposes less of penalty compared with the ability to use information in real-time. [ABSTRACT FROM PUBLISHER]

Linear programming (LP) decoding for low-density parity-check codes was introduced by Feldman et al. and has been shown to have theoretical guarantees in several regimes. Furthermore, it has been reported in the literature—via simulation and via instanton analysis—that LP decoding displays better error rate performance at high signal-to-noise ratios (SNR) than does belief propagation (BP) decoding. However, at low SNRs, LP decoding is observed to have worse performance than BP. In this paper, we seek to improve LP decoding at low SNRs while maintaining LP decoding’s high SNR performance. Our main contribution is a new class of decoders obtained by applying the alternating direction method of multipliers (ADMM) algorithm to a set of non-convex optimization problems. These non-convex problems are constructed by adding a penalty term to the objective of LP decoding. The goal of the penalty is to make pseudocodewords, which are non-integer vertices of the LP relaxation, more costly. We name this class of decoders—ADMM penalized decoders. For low and moderate SNRs, we simulate ADMM penalized decoding with \ell 1 and \ell 2 penalties. We find that these decoders can outperform both BP and LP decoding. For high SNRs, where it is difficult to obtain data via simulation, we use an instanton analysis and find that, asymptotically, ADMM penalized decoding performs better than BP but not as well as LP. Unfortunately, since ADMM penalized decoding is not a convex program, we have not been successful in developing theoretical guarantees. However, the non-convex program can be approximated using a sequence of linear programs; an approach that yields a reweighted LP decoder. We show that a two-round reweighted LP decoder has an improved theoretical recovery threshold when compared with LP decoding. In addition, we find via simulation that reweighted LP decoding significantly attains lower error rates than LP decoding at low SNRs. [ABSTRACT FROM AUTHOR]

This paper investigates various subtleties of applying linear physical-layer network coding (PNC) with $q$ -level pulse amplitude modulation ( $q$ -PAM) in two-way relay channels. A critical issue is how the PNC system performs when the received powers from the two users at the relay are imbalanced. In particular, how would the PNC system perform under slight power imbalance that is inevitable in practice, even when power control is applied? To answer these questions, this paper presents a comprehensive analysis of $q$ -PAM PNC. Our contributions are as follows. First, we give a systematic way to obtain the analytical relationship between the minimum distance of the signal constellation induced by the superimposed signals of the two users (a key performance determining factor) and the channel-gain ratio of the two users, for all $q$ . In particular, we show how the minimum distance changes in a piecewise linear fashion as the channel-gain ratio varies. Second, we show that the performance of $q$ -PAM PNC is highly sensitive to imbalanced received powers from the two users at the relay, even when the power imbalance is slight (e.g., the residual power imbalance in a power-controlled system). This sensitivity problem is exacerbated as $q$ increases, calling into question the robustness of high-order modulated PNC. Third, we propose an asynchronized PNC system in which the symbol arrival times of the two users at the relay are deliberately made to be asynchronous. We show that such asynchronized PNC, when operated with a belief propagation decoder, can remove the sensitivity problem, allowing a robust high-order modulated PNC system to be built. [ABSTRACT FROM AUTHOR]

This paper presents an outage analysis for a two-user parallel Gaussian interference channel consisting of two sub-channels. Each sub-channel is modeled as a two-user Gaussian interference channel with quasi-static and flat fading. Both users employ single-layer Gaussian code-books and maintain a statistical correlation \rho between the signals transmitted over the underlying sub-channels. When joint decoding (JD) is performed at the receivers, setting \rho =0 minimizes the outage probability, regardless of the value of the signal-to-noise ratio (SNR). It is shown, however, that if the receivers treat interference as noise (TIN) or cancel interference (CI), the value of optimum \rho approaches 1 as SNR goes to infinity. Motivated by these observations, we let \rho =0 under JD and \rho =1 , it is shown that the outage probability scales like \mathrm snr^-(1-r) under both TIN and CI, while it vanishes at least as fast as \mathrm snr^-\min \2-r,4(1-r)\\log \mathrm snr under JD. This paper is concluded by extending some of the results to a two-user parallel Gaussian interference channel with an arbitrary number of sub-channels. [ABSTRACT FROM AUTHOR]

We propose two coding schemes for the two-receiver discrete memoryless broadcast channel (BC) with rate-limited feedback from one or both receivers. They improve over the no feedback capacity region for a large class of channels, including the class of strictly essentially less-noisy BCs that we introduce in this paper. Examples of strictly essentially less-noisy BCs are the binary symmetric BC or the binary erasure BC with unequal crossover or erasure probabilities at the two receivers. When the feedback rates are sufficiently large, our schemes recover all previously known capacity results for discrete memoryless BCs with feedback. In both our schemes, we let the receivers feedback quantization messages about their receive signals. In the first scheme, the transmitter simply relays the quantization information obtained from Receiver 1 to Receiver 2, and vice versa. This provides each receiver with a second observation of the input signal and can thus improve its decoding performance unless the BC is physically degraded. Moreover, each receiver uses its knowledge of the quantization message describing its own outputs so as to attain the same performance as if this message had not been transmitted at all. In our second scheme, the transmitter first reconstructs and processes the quantized output signals, and then sends the outcome as a common update information to both receivers. A special case of our second scheme also applies to memoryless BCs without feedback but with strictly causal state-information at the transmitter and causal state-information at the receivers. It recovers all previous achievable regions also for this setup with state-information. [ABSTRACT FROM AUTHOR]

Gabidulin codes can be seen as the rank-metric equivalent of Reed–Solomon codes. It was recently proved, using subspace polynomials, that Gabidulin codes cannot be list decoded beyond the so-called Johnson radius. In another result, cyclic subspace codes were constructed by inspecting the connection between subspaces and their subspace polynomials. In this paper, these subspace codes are used to prove two bounds on the list size in decoding certain Gabidulin codes. The first bound is an existential one, showing that exponentially sized lists exist for codes with specific parameters. The second bound presents exponentially sized lists explicitly for a different set of parameters. Both bounds rule out the possibility of efficiently list decoding several families of Gabidulin codes for any radius beyond half the minimum distance. Such a result was known so far only for non-linear rank-metric codes, and not for Gabidulin codes. Using a standard operation called lifting, identical results also follow for an important class of constant dimension subspace codes. [ABSTRACT FROM AUTHOR]

Subspace codes have received an increasing interest recently due to their application in error correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and decoding algorithms. In this paper, we consider such cyclic codes and provide constructions of optimal codes for which their codewords do not have full orbits. We further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials. We present some constructions of such codes, which are cyclic and analyze their parameters. [ABSTRACT FROM AUTHOR]

Polar coding is a method for communication over noisy classical channels, which is provably capacity achieving and has an efficient encoding and decoding. Recently, this method has been generalized to the realm of quantum information processing, for tasks such as classical communication, private classical communication, and quantum communication. In this paper, we apply the polar coding method to network classical-quantum information theory, by making use of recent advances for related classical tasks. In particular, we consider problems such as the compound multiple access channel and the quantum interference channel. The main result of our work is that it is possible to achieve the best known inner bounds on the achievable rate regions for these tasks, without requiring a so-called quantum simultaneous decoder. Thus, this paper paves the way for developing network classical-quantum information theory further without requiring a quantum simultaneous decoder. [ABSTRACT FROM AUTHOR]

Burst errors are a type of distortion in many data communications and data storage channels. In this paper, we consider the list decodability of codes for single burst error case and phased-burst error case independently. Firstly, we analyze the list decodability of random codes, and we show that the burst list decoding radius and the rate of random codes achieve the Singleton bound and the Gilbert-Varshamov bound for single case and phased case, respectively. Second, we illustrate that cyclic codes and algebraic geometry codes are good burst list-decodable codes for single case and phased case, respectively. [ABSTRACT FROM AUTHOR]

In this paper, the multiple access channel (MAC) with combined cooperation and partial cribbing is studied, and its capacity region is characterized. Cooperation means that each of the two encoders sends a message to the other via a rate-limited link prior to transmission, while partial cribbing means that each of the two encoders obtains a deterministic function of the other encoder’s output with or without delay. Prior work in this field dealt separately with cooperation and partial cribbing, but by combining these two methods, we can achieve significantly higher rates. Surprisingly, the capacity region of the MAC with combined cooperation and partial cribbing can be expressed using only one auxiliary random variable (RV) similar to the capacity regions of the MAC with cooperation and with partially cribbing encoders. The reason is that in an optimal coding scheme, the encoders use both cooperation and partial cribbing to generate a common message between the encoders. Furthermore, the Gaussian MAC with combined one-sided cooperation and quantized cribbing is studied. For this model, an achievability scheme is given. This scheme shows how many cooperation or quantization bits are required to practically achieve the capacity region of the Gaussian MAC with full message cooperation or perfect cribbing. To ratify the main results, two additional models are studied. In both models, only one auxiliary RV is needed. The first is a rate distortion dual setting for the MAC with degraded message set and combined cooperation and cribbing. The second is a state-dependent MAC with cooperation, where the state is known at a partially cribbing encoder and at the decoder. However, there are cases where more than one auxiliary RV is needed, e.g., when the cooperation and the cribbing are not used for the same purposes. The MAC with an action-dependent state is presented, where the action is based on the cooperation but not on the cribbing. Therefore, in this case, more than one auxiliary RV is needed. As a result, when the common information shared by the two encoders is used unevenly by the users in the channel, more than one auxiliary RV is needed to express the capacity region. [ABSTRACT FROM AUTHOR]

An upper bound on the error probability of specific lattices, based on their distance spectrum, is constructed. The derivation is accomplished using a simple alternative to the Minkowski–Hlawka mean-value theorem of the geometry of numbers. In many ways, the new bound greatly resembles the Shulman–Feder bound for linear codes. Based on the new bound, error-exponent and channel-dispersion expressions are derived for specific lattice sequences (of increasing dimension) over the AWGN channel. Measuring a sequence’s gap to capacity, using the new asymptotics, is demonstrated. Additional finite dimension results, encountered along the way, are presented. [ABSTRACT FROM PUBLISHER]

Despite the simplicity of the scheme of treating interference as noise (TIN), it was shown to be sum-capacity optimal in the Gaussian interference channel (IC) with very-weak (noisy) interference. In this paper, the two-user IC is altered by introducing an additional transmitter that wants to communicate with one of the receivers of the IC. The resulting network thus consists of a point-to-point channel interfering with a multiple access channel (MAC) and is denoted by PIMAC. The sum-capacity of the PIMAC is studied with main focus on the optimality of TIN. It turns out that TIN in its naive variant, where all transmitters are active and both receivers use TIN for decoding, is not the best choice for the PIMAC. In fact, a scheme that combines both time division multiple access and TIN (TDMA–TIN) strictly outperforms the naive-TIN scheme. Furthermore, it is shown that in some regimes, TDMA–TIN achieves the sum-capacity for the deterministic PIMAC and the sum-capacity within a constant gap for the Gaussian PIMAC. In addition, it is shown that, even for very-weak interference, there are some regimes where a combination of interference alignment with power control and TIN at the receiver side outperforms TDMA–TIN. As a consequence, on the one hand, TIN in a cellular uplink is approximately optimal in certain regimes. On the other hand, those regimes cannot be simply described by the strength of interference. [ABSTRACT FROM PUBLISHER]

In this paper, a random access scheme is introduced, which relies on the combination of packet erasure correcting codes and successive interference cancellation (SIC). The scheme is named coded slotted ALOHA. A bipartite graph representation of the SIC process, resembling iterative decoding of generalized low-density parity-check codes over the erasure channel, is exploited to optimize the selection probabilities of the component erasure correcting codes through a density evolution analysis. The capacity (in packets per slot) of the scheme is then analyzed in the context of the collision channel without feedback. Moreover, a capacity bound is developed, and component code distributions tightly approaching the bound are derived. [ABSTRACT FROM AUTHOR]