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In an interesting paper, Prof. C. Ding provided three constructions of cyclic codes of length being a product of two primes. Numerical data shows that many codes from these constructions are best cyclic codes of the same length and dimension over the same finite field. However, not much is known about these codes. In this paper, we explain some of the numerical data by developing a general method on cyclic codes of composite length and on estimating the minimum distance. We also provide a general construction of cyclic codes of composite length which are related to Ding’s constructions. Numerical data shows that it produces many best cyclic codes as well. Finally, we point out how these cyclic codes can be used to construct convolutional codes with large free distance. [ABSTRACT FROM AUTHOR]

The zero-error capacity of a channel is defined as the maximum rate at which it is possible to transmit information with zero probability of error. In this paper, we settle all previously unsolved cases for the zero-error capacity of binary channels with one memory. [ABSTRACT FROM AUTHOR]

In the $q$ -ary online (or “causal”) channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf {x} =(x_{1},\ldots,x_{n}) \in \{0,1,\ldots,q-1\}^{n}$ symbol-by-symbol via a channel limited to at most $pn$ errors and $p^{*} n$ erasures. The channel is “online” in the sense that at the $i$ th step of communication the channel decides whether to corrupt the $i$ th symbol or not based on its view so far, i.e., its decision depends only on the transmitted symbols $(x_{1},\ldots,x_{i})$. This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has full knowledge of the sent codeword $\mathbf {x}$. In this paper, we study the capacity of $q$ -ary online channels for a combined corruption model, in which the channel may impose at most $pn$ errors and at most $p^{*} n$ erasures on the transmitted codeword. The online channel (in both the error and erasure case) has seen a number of recent studies, which present both upper and lower bounds on its capacity. In this paper, we give a full characterization of the capacity as a function of $q,p$ , and $p^{*}$. [ABSTRACT FROM AUTHOR]

This paper generalizes the method proposed by Poulliat et al. for the determination of the optimal Galois field coefficients of a non-binary LDPC parity check constraint based on the binary image of the code. Optimal, or almost-optimal, parity check coefficients are given for check degree varying from 4 to 20 and Galois field varying from GF (64) up to GF (1024). For all given sets of coefficients, no codeword of Hamming weight two exists. A reduced complexity algorithm to compute the binary Hamming weight 3 of a parity check is proposed. When the number of sets of coefficients is too high for an exhaustive search and evaluation, a local greedy search is performed. Explicit tables of coefficients are given. The proposed sets of coefficients can effectively replace the random selection of coefficients often used in NB-LDPC construction. [ABSTRACT FROM AUTHOR]

Splitter sets are closely related to lattice tilings, and have applications in flash memories and conflict avoiding codes. In this paper, we prove some nonexistence results for nonsingular perfect splitter sets. We also give some necessary conditions for the existence of purely singular perfect splitter sets. Finally, we apply these results to purely singular perfect $B[-1,k](m)$ and $B[-2,k](m)$ sets for small $k$. In particular, we solve completely the problems left by Schwartz (European J. Combin., vol. 36, pp.130–142, Feb. 2014). [ABSTRACT FROM AUTHOR]

We consider list decoding in the zero-rate regime for two cases—the binary alphabet and the spherical codes in Euclidean space. Specifically, we study the maximal $\tau \in [{0,1}]$ for which there exists an arrangement of $M$ balls of relative Hamming radius $\tau $ in the binary hypercube (of arbitrary dimension) with the property that no point of the latter is covered by $L$ or more of them. As $M\to \infty $ the maximal $\tau $ decreases to a well-known critical value $\tau _{L}$. In this paper, we prove several results on the rate of this convergence. For the binary case, we show that the rate is $\Theta (M^{-1})$ when $L$ is even, thus extending the classical results of Plotkin and Levenshtein for $L=2$. For $L=3$ , the rate is shown to be $\Theta (M^{-({2}/{3})})$. For the similar question about spherical codes, we prove the rate is $\Omega (M^{-1})$ and $O(M^{-({2L}/{L^{2}-L+2})})$. [ABSTRACT FROM AUTHOR]

Estimating mutual information from independent identically distributed samples drawn from an unknown joint density function is a basic statistical problem of broad interest with multitudinous applications. The most popular estimator is the one proposed by Kraskov, Stögbauer, and Grassberger (KSG) in 2004 and is nonparametric and based on the distances of each sample to its $k^{\mathrm{ th}}$ nearest neighboring sample, where $k$ is a fixed small integer. Despite of its widespread use (part of scientific software packages), theoretical properties of this estimator have been largely unexplored. In this paper, we demonstrate that the estimator is consistent and also identify an upper bound on the rate of convergence of the $\ell _{2}$ error as a function of a number of samples. We argue that the performance benefits of the KSG estimator stems from a curious “correlation boosting” effect and build on this intuition to modify the KSG estimator in novel ways to construct a superior estimator. As a by-product of our investigations, we obtain nearly tight rates of convergence of the $\ell _{2}$ error of the well-known fixed $k$ -nearest neighbor estimator of differential entropy by Kozachenko and Leonenko. [ABSTRACT FROM AUTHOR]

Codes that can correct up to $t$ symmetric errors and detect all unidirectional errors, known as $t$ -EC-AUED codes, are studied in this paper. Given positive integers $q$ , $a$ , and $t$ , let $n_{q}(a,t+1)$ denote the length of the shortest $q$ -ary $t$ -EC-AUED code of size $a$. We introduce combinatorial constructions for $q$ -ary $t$ -EC-AUED codes via one-factorizations of complete graphs, and concatenation of MDS codes and codes from resolvable set systems. Consequently, we determine the exact values of $n_{q}(a,t+1)$ for several new infinite families of $q,a$ , and $t$. [ABSTRACT FROM AUTHOR]

In this paper, new index coding problems are studied, where each receiver has erroneous side information. Although side information is a crucial part of index coding, the existence of erroneous side information has not been considered yet. We study an index code with receivers that have erroneous side information symbols in the error-free broadcast channel, which is called an index code with side information errors (ICSIE). The encoding and decoding procedures of the ICSIE are proposed, based on the syndrome decoding. Then, we derive the bounds on the optimal codelength of the proposed index code with erroneous side information. Furthermore, we introduce a special graph for the proposed index coding problem, called a \delta s -cycle whose properties are similar to those of the cycle in the conventional index coding problem. Properties of the ICSIE are also discussed in the \delta s -cycle and clique. Finally, the proposed ICSIE is generalized to an index code for the scenario having both additive channel errors and side information errors, called a generalized error correcting index code. [ABSTRACT FROM AUTHOR]

Low rank approximation is an important tool used in many applications of signal processing and machine learning. Recently, randomized sketching algorithms were proposed to effectively construct low rank approximations and obtain approximate singular value decompositions of large matrices. Similar ideas were used to solve least squares regression problems. In this paper, we show how matrices from error correcting codes can be used to find such low rank approximations and matrix decompositions, and extend the framework to linear least squares regression problems. The benefits of using these code matrices are the following. First, they are easy to generate and they reduce randomness significantly. Second, code matrices, with mild restrictions, satisfy the subspace embedding property, and have a better chance of preserving the geometry of an large subspace of vectors. Third, for parallel and distributed applications, code matrices have significant advantages over structured random matrices and Gaussian random matrices. Fourth, unlike Fourier or Hadamard transform matrices, which require sampling O(k\log k) columns for a rank- k approximation, the log factor is not necessary for certain types of code matrices. In particular, (1+ \epsilon) optimal Frobenius norm error can be achieved for a rank- k approximation with O(k/ \epsilon) where A\in \mathbb R ^n\times d , the (1+ \epsilon) relative error approximation can be achieved with $O(d/ \epsilon)$ samples, with high probability, when certain code matrices are used. [ABSTRACT FROM AUTHOR]

Secure network coding assumes that the underlying network channels are error-free; thus, if our channels introduce errors, we need to first apply a channel code to correct them, and then build security on top of the resulting error-free network. In this paper, we develop achievability protocols and outer bounds for the secure network coding setting, where the edges are subject to packet erasures, and public feedback of the channel state is available to both Eve and the legitimate network nodes. We show that by leveraging erasures and feedback, we can achieve secrecy rates that are in some cases multiple times higher than the alternative of separate channel-error-correction followed by secure network coding; moreover, we develop outer bounds and prove optimality of our proposed schemes in some special cases. [ABSTRACT FROM AUTHOR]

In this paper, we study the graphical structure of elementary trapping sets (ETSs) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree \(d_{l}\) and girth \(g\) , we identify all the nonisomorphic structures of an arbitrary class of \((a,b)\) ETSs, where \(a\) is the number of variable nodes and \(b\) is the number of odd-degree check nodes in the induced subgraph of the ETS. This paper leads to a simple characterization of dominant classes of ETSs (those with relatively small values of \(a\) and \(b\) ) based on short cycles in the Tanner graph of the code. For such classes of ETSs, we prove that any set \({\cal S}\) in the class is a layered superset (LSS) of a short cycle, where the term layered is used to indicate that there is a nested sequence of ETSs that starts from the cycle and grows, one variable node at a time, to generate \({\cal S}\) . This characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the ETSs with LSS property in a guaranteed fashion. Specific results on the structure of ETSs are presented for \(d_{l} = 3, 4, 5, 6\) , \(g = 6, 8\) , and \(a, b \leq 10\) in this paper. The results of this paper can be used for the error floor analysis and for the design of LDPC codes with low error floors. [ABSTRACT FROM AUTHOR]

We introduce complementary information set codes of higher order. A binary linear code of length \(tk\) and dimension \(k\) is called a complementary information set code of order \(t\) ( \(t\) -CIS code for short) if it has \(t\) pairwise disjoint information sets. The duals of such codes permit to reduce the cost of masking cryptographic algorithms against side-channel attacks. As in the case of codes for error correction, given the length and the dimension of a \(t\) -CIS code, we look for the highest possible minimum distance. In this paper, this new class of codes is investigated. The existence of good long CIS codes of order 3 is derived by a counting argument. General constructions based on cyclic and quasi-cyclic codes and on the building up construction are given. A formula similar to a mass formula is given. A classification of 3-CIS codes of length \(\le 12\) is given. Nonlinear codes better than linear codes are derived by taking binary images of \( {\mathbb Z}_{4}\) -codes. A general algorithm based on Edmonds’ basis packing algorithm from matroid theory is developed with the following property: given a binary linear code of rate \(1/t\) , it either provides \(t\) disjoint information sets or proves that the code is not \(t\) -CIS. Using this algorithm, all optimal or best known \([tk, k]\) codes, where \(t=3, 4, {\dots }, 256\) and \(1 \le k \le \lfloor 256/t \rfloor \) are shown to be \(t\) -CIS for all such \(k\) and \(t\) , except for \(t=3\) with \(k=44\) and \(t=4\) with \(k=37\) . [ABSTRACT FROM AUTHOR]

We consider the decoding of linear and array codes from errors when we are only allowed to download a part of the codeword. More specifically, suppose that we have encoded $k$ data symbols using an $(n,k)$ code with code length $n$ and dimension $k$. During storage, some of the codeword coordinates might be corrupted by errors. We aim to recover the original data by reading the corrupted codeword with a limit on the transmission bandwidth, namely, we can only download an $\alpha $ proportion of the corrupted codeword. For a given $\alpha $ , our objective is to design a code and a decoding scheme such that we can recover the original data from the largest possible number of errors. A naive scheme is to read $\alpha n$ coordinates of the codeword. This method used in conjunction with MDS codes guarantees recovery from any $\lfloor (\alpha n-k)/2\rfloor $ errors. In this paper we show that we can instead download an $\alpha $ proportion from each of the codeword’s coordinates. For a well-designed MDS code, this method can guarantee recovery from $\lfloor (n-k/\alpha)/2 \rfloor $ errors, which is $1/\alpha $ times more than the naive method, and is also the maximum number of errors that an $(n,k)$ code can correct by downloading only an $\alpha $ proportion of the codeword. We present two families of such optimal constructions and decoding schemes of which one is based on Interleaved Reed-Solomon codes and the other on Folded Reed-Solomon codes. We further show that both code constructions attain asymptotically optimal list decoding radius when downloading only a part of the corrupted codeword. We also construct an ensemble of random codes that with high probability approaches the upper bound on the number of correctable errors when the decoder downloads an $\alpha $ proportion of the corrupted codeword. [ABSTRACT FROM AUTHOR]

The Z2s-additive codes are subgroups of Zn2s, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z4-linear Hadamard codes. However, when s > 2, the dimension of the kernel of Z2s-linear Hadamard codes of length 2t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is computed for s = 3. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once t ≥ 3 is fixed. In this case, the number of nonequivalent such codes is also established. [ABSTRACT FROM AUTHOR]

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

We provide polylog sparse quantum codes for correcting the Erasure channel arbitrarily close to the capacity. Specifically, we provide the $[[n, k, d]]$ quantum stabilizer codes that correct for the erasure channel arbitrarily close to the capacity if the erasure probability is at least 0.33 and with a generating set $\langle S_{1}, S_{2}, \cdots {} S_{n-k} \rangle $ such that $|S_{i}|\leq {\mathrm{ log}} ^{2+\zeta }(n)$ for all $i$ and for any $\zeta > 0$ with high probability. In this paper, we show that the result of Delfosse et al. is tight: one can construct capacity approaching codes with weight almost $O(1)$. [ABSTRACT FROM AUTHOR]

In this paper, we first study deviations of the complete weight distribution of a linear code from that of a random code. Then, we consider a large family of subfield subcodes of algebraic-geometric codes over prime fields which include BCH codes and Goppa codes and prove that the complete weight distribution is close to that of a random code if the code length is large compared with the genus of the curve and the degree of the divisor defining the code. [ABSTRACT FROM AUTHOR]

We use deterministic and probabilistic methods to analyze the performance of compressed sensing matrices constructed from Hadamard matrices and pairwise balanced designs, previously introduced by a subset of the authors. In this paper, we obtain upper and lower bounds on the sparsity of signals for which our matrices guarantee recovery. These bounds are tight to within a multiplicative factor of at most 4\sqrt 2 . We provide new theoretical results and detailed simulations, which indicate that the construction is competitive with Gaussian random matrices, and that recovery is tolerant to noise. A new recovery algorithm tailored to the construction is also given. [ABSTRACT FROM AUTHOR]

Regenerating code is a class of distributed storage codes that can optimally trade the bandwidth required to repair a failed node with the amount of data stored per node. There are two optimal points in the regeneration tradeoff curve: the minimum storage regeneration code and the minimum bandwidth regeneration code. However, in hostile networks where the storage nodes may be compromised, the storage capacity of the network can be significantly affected. In this paper, we propose two optimal regenerating code constructions through rate-matching to combat this kind of adversarial attacks in hostile networks. We first develop a two-layer rate-matched regenerating code construction. By matching the parameters of the full rate code and the partial rate code, we can optimize the overall storage efficiency while maintaining the corrupted node detection probability. Through comprehensive analysis, we show that the two-layer rate-matched regenerating code can achieve 70% higher storage efficiency than the universally resilient regenerating code. We then propose an optimal $m$ -layer regenerating code construction. While the principle remains the same as the two-layer code, it is designed to optimize the total number of detectable corrupted nodes of $m$ layers from which the errors can be corrected under the constraint of any given code efficiency. Compared with the universally resilient regenerating code with the same rate, our $m$ -layer code can detect 50% more corrupted nodes. [ABSTRACT FROM PUBLISHER]

The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst case deletions, with a focus on constructing efficiently decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following tradeoffs (for any \varepsilon > 0 ): 1) codes that can correct a fraction 1- \varepsilon of deletions with rate \mathop \mathrm poly\nolimits ( \varepsilon ) over an alphabet of size \mathop \mathrm poly\nolimits (1/ \varepsilon ) ; 2) binary codes of rate 1-\tilde O(\sqrt \varepsilon ) that can correct a fraction $ \varepsilon $ of deletions; and 3) Binary codes that can be list-decoded from a fraction (1/2- \varepsilon )$ of deletions with rate \mathop {\mathrm {poly}}\nolimits ( \varepsilon ) . This paper gives the first efficient constructions which meet the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets, and correcting a constant fraction of bit deletions with rate approaching 1 over a fixed alphabet. The above-mentioned results bring our understanding of deletion code constructions in these regimes to a similar level as worst case errors. [ABSTRACT FROM PUBLISHER]

In this paper, list decoding of crisscross errors in arrays over finite fields is considered. For this purpose, the so-called cover metric is used, where the cover of a matrix is a set of rows and columns which contains all non-zero elements of the matrix. A Johnson-like upper bound on the maximum list size in the cover metric is derived, showing that the list of codewords has polynomial size up to a certain radius. Furthermore, a simple list decoding algorithm for a known optimal code construction is presented, which decodes errors in the cover metric up to our upper bound. These results reveal significant differences between the cover metric and the rank metric and show that the cover metric is more suitable for correcting crisscross errors. [ABSTRACT FROM PUBLISHER]

One central theme in quantum error-correction is to construct quantum codes that have a relatively large minimum distance. In this paper, we first present a construction of classical linear codes based on certain classes of polynomials. Through these classical linear codes, we are able to obtain some new quantum codes. It turns out that some of the quantum codes exhibited here have better parameters than the ones available in the literature. [ABSTRACT FROM PUBLISHER]

Grassmannian codes are known to be useful in error correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet size. The multicast networks which were used for this purpose are generalized combination networks. In both the scalar and the vector network coding solutions, the subspace distance is used as the distance measure for the codes which solve the network coding problem in the generalized combination networks. In this paper, we show that the subspace distance can be replaced with two other possible distance measures which generalize the subspace distance. These two distance measures are shown to be equivalent under an orthogonal transformation. It is proved that the Grassmannian codes with the new distance measures generalize the Grassmannian codes with the subspace distance and the subspace designs with the strength of the design. Furthermore, optimal Grassmannian codes with the new distance measures have minimal requirements for the network coding solutions of some generalized combination networks. The coding problems related to these two distance measures, especially with respect to network coding, are discussed. Finally, by using these new concepts, it is proved that the codes in the Hamming scheme form a subfamily of the Grassmannian codes. [ABSTRACT FROM AUTHOR]

In recent years, self-dual MDS codes have attracted a lot of attention due to theoretical interest and practical importance. Similar to self-dual MDS codes, self-dual near MDS (NMDS for short) codes have nice structures as well. From both theoretical and practical points of view, it is natural to study self-dual NMDS codes. Although there has been lots of work on NMDS codes in literature, little is known for self-dual NMDS codes. It seems more challenging to construct self-dual NMDS codes than self-dual MDS codes. The only work on construction of self-dual NMDS codes shows existence of $q$ -ary self-dual NMDS codes of length $q-1$ for odd prime power $q$ or length up to 16 for some small primes $q$ with $q\le 197$. In this paper, we make use of properties of elliptic curves to construct self-dual NMDS codes. It turns out that, as long as $2|q$ and $n$ is even with $4\le n\le q+\lfloor 2\sqrt {q}\rfloor -2$ , one can construct a self-dual NMDS code of length $n$ over $ {\mathbb {F}}_{q}$. Furthermore, for odd prime power $q$ , there exists a self-dual NMDS code of length $n$ over $ {\mathbb {F}}_{q}$ if $q\ge 4^{n+3}\times (n+3)^{2}$. [ABSTRACT FROM AUTHOR]

Cages, defined as regular graphs with minimum number of nodes for a given girth, are well-studied in graph theory. Trapping sets are graphical structures responsible for error floor of low-density parity-check (LDPC) codes, and are well investigated in coding theory. In this paper, we make connections between cages and trapping sets. In particular, starting from a cage (or a modified cage), we construct a trapping set in multiple steps. Based on the connection between cages and trapping sets, we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable-regular LDPC codes with a given variable degree and girth. The derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets. Considering that non-zero codewords are a special case of trapping sets, we also derive tight upper bounds on the minimum weight of such codewords, i.e., the minimum distance, of variable-regular LDPC codes as a function of variable degree and girth. [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]

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]

In network communication, the source often transmits messages at several different information rates within a session. How to deal with information transmission and network error correction simultaneously under different rates is introduced in this paper as a variable-rate network error correction problem. Apparently, linear network error correction maximum distance separable (MDS) codes are expected to be used for these different rates guaranteeing the maximal error-correcting capability. For this purpose, designing a linear network error correction MDS code based on the existing results for each information rate is an alternative solution, but it is inefficient due to its high complexity. In order to solve the problem more efficiently, we present the concept of variable-rate linear network error correction MDS codes preserving local encoding kernels, that is, these linear network error correction MDS codes of different rates have the same local encoding kernel at each internal node. Thus, each nonsource node always uses the same local kernel for coding, no matter what the rate is. Furthermore, we propose an approach to construct such a family of variable-rate network MDS codes and give an algorithm for efficient implementation. This approach economizes the storage space for each internal node, and saves resources and time for transmissions on networks. Moreover, the performance of our proposed algorithm is analyzed, including the field size, the time complexity, the encoding complexity at the source node, and the decoding methods. Finally, a random method is introduced for constructing such a family of variable-rate network MDS codes, and a lower bound on the success probability of this random method is given, which shows that this probability will approach to one as the base field size goes to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new way of constructing and decoding multipermutation codes. Multipermutations are the permutations of a multiset that generally consist of duplicate entries. We first introduce a class of binary matrices called multipermutation matrices, each of which corresponds to a unique and distinct multipermutation. By enforcing a set of linear constraints on these matrices, we define a new class of codes that we term linear program (LP)-decodable multipermutation codes. In order to decode these codes using an LP, thereby enabling soft decoding, we characterize the convex hull of multipermutation matrices. This characterization allows us to relax the coding constraints to a polytope and to derive two LP decoding problems. These two problems are, respectively, formulated by relaxing the maximum likelihood decoding problem and the minimum Chebyshev distance decoding problem. Because these codes are non-linear, we also study efficient encoding and decoding algorithms. We first describe an algorithm that maps consecutive integers, one by one, to an ordered list of multipermutations. Based on this algorithm, we develop an encoding algorithm for a code proposed by Shieh and Tsai, a code that falls into our class of LP-decodable multipermutation codes. Regarding decoding algorithms, we propose an efficient distributed decoding algorithm based on the alternating direction method of multipliers. Finally, we observe from the simulation results that the soft decoding techniques we introduce can significantly outperform hard decoding techniques that are based on quantized channel outputs. [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]

Maximum distance separable (MDS) block codes and MDS 1D convolutional codes are the most robust codes for error correction within the class of block codes of a fixed rate and 1D convolutional codes of a certain rate and degree, respectively. In this paper, we generalize this concept to the class of 2D convolutional codes. For that, we introduce a natural bound on the distance of a 2D convolutional code of rate k/n and degree \delta , which generalizes the Singleton bound for block codes and the generalized Singleton bound for 1D convolutional codes. Then, we prove the existence of 2D convolutional codes of rate k/n and degree \delta if k {\nmid } \delta , or n \geq k (({(({\delta }/{k}) + 1)(({\delta }/{k}) + 2)})/{2})$ if $k \mid \delta $ , by presenting a concrete constructive procedure. [ABSTRACT FROM AUTHOR]

In this paper, we obtain performance guarantees for modified-CS and for its improved version, modified-CS-Add-LS-Del, for recursive reconstruction of a time sequence of sparse signals from a reduced set of noisy measurements available at each time. Under mild assumptions, we show that the support recovery error of both algorithms is bounded by a time-invariant and small value at all times. The same is also true for the reconstruction error. Under a slow support change assumption: 1) the support recovery error bound is small compared with the support size and 2) our results hold under weaker assumptions on the number of measurements than what \ell 1 minimization for noisy data needs. We first give a general result that only assumes a bound on support size, number of support changes, and number of small magnitude nonzero entries at each time. Later, we specialize the main idea of these results for two sets of signal change assumptions that model the class of problems in which a new element that is added to the support either gets added at a large initial magnitude or its magnitude slowly increases to a large enough value within a finite delay. Simulation experiments are shown to back up our claims. [ABSTRACT FROM PUBLISHER]

One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate $R$ and minimum distance $\delta $ of a binary code. The best known tradeoff is the Gilbert–Varshamov bound, and says that for every $\delta \in (0, 1/2)$ , there are codes with minimum distance $\delta $ and rate $R = {R_{\mathsf {GV}}}(\delta) > 0$ (for a certain simple function ${R_{\mathsf {GV}}}(\cdot)$ ). In this paper, we show that the Gilbert–Varshamov bound can be achieved by codes, which support local error-detection and error-correction algorithms. Specifically, we show the following results. 1) Local testing: for all $\delta \in (0,1/2)$ and all $R < {R_{\mathsf {GV}}}(\delta)$ , there exist codes with length $n$ , rate $R$ , and minimum distance $\delta $ that are locally testable with $\mathsf {quasipolylog}(n)$ query complexity. 2) Local correction: for all $\epsilon > 0$ , for all $\delta < 1/2$ sufficiently large, and all $R < (1-\epsilon) {R_{\mathsf {GV}}}(\delta)$ , there exist codes with length $n$ , rate $R$ , and minimum distance $\delta $ that are locally correctable from $({\delta }/{2}) - o(1)$ fraction errors with $O(n^{\epsilon })$ query complexity. Furthermore, these codes have an efficient randomized construction, and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity. Our results on locally correctable codes also immediately give locally decodable codes with the same parameters. Our local testing result is obtained by combining Thommesen’s random concatenation technique and the best known locally testable codes by Kopparty et al. Our local correction result, which is significantly more involved, also uses random concatenation, along with a number of further ideas: the Guruswami–Sudan–Indyk list decoding strategy for concatenated codes, Alon–Edmonds–Luby distance amplification, and the local list-decodability, local list-recoverability, and local testability of Reed–Muller codes. Curiously, our final local correction algorithms go via local list-decoding and local testing algorithms; this seems to be the first time local testability is used in the construction of a locally correctable code. [ABSTRACT FROM AUTHOR]

In this paper, we improve the layered implementation of arbitrary stabilizer circuits introduced by Aaronson and Gottesman in Phys. Rev. A 70 (052328), 2004: to implement a general stabilizer circuit, we reduce their 11-stage computation -H-C-P-C-P-C-H-P-C-P-C- over the gate set consisting of Hadamard, controlled-NOT, and phase gates, into a 7-stage computation of the form -C-CZ-P-H-P-CZ-C-. We show arguments in support of using -CZ- stages over the -C- stages: not only the use of -CZ- stages allows a shorter layered expression, but -CZ- stages are simpler and appear to be easier to implement compared to the -C- stages. Based on this decomposition, we develop a two-qubit gate depth- $(14n{-}4)$ implementation of stabilizer circuits over the gate library $\{ \mathrm{H}, \mathrm{P}, \mathrm{CNOT}\}$ , executable in the Linear Nearest Neighbor (LNN) architecture, improving best previously known depth- $25n$ circuit, also executable in the LNN architecture. Our constructions rely on Bruhat decomposition of the symplectic group and on folding arbitrarily long sequences of the form (-P-C-) $^{m}$ into a three-stage computation -P-CZ-C-. Our results include the reduction of the 11-stage decomposition -H-C-P-C-P-C-H-P-C-P-C- into a 9-stage decomposition of the form -C-P-C-P-H-C-P-C-P-. This reduction is based on the Bruhat decomposition of the symplectic group. This result also implies a new normal form for stabilizer circuits. We show that a circuit in this normal form is optimal in the number of Hadamard gates used. We also show that the normal form has an asymptotically optimal number of parameters. [ABSTRACT FROM AUTHOR]

This paper reduces multiple-unicast network coding to single-unicast secure network coding and single-unicast network error correction. Specifically, we present reductions that map an arbitrary multiple-unicast network coding instance to a unicast secure network coding instance in which at most one link is eavesdropped, or a unicast network error correction instance in which at most one link is erroneous, such that a rate tuple is achievable in the multiple-unicast network coding instance if and only if a corresponding rate is achievable in the unicast secure network coding instance, or in the unicast network error correction instance. Conversely, we show that an arbitrary unicast secure network coding instance in which at most one link is eavesdropped can be reduced back to a multiple-unicast network coding instance. In addition, we show that the capacity of a unicast network error correction instance in general is not (exactly) achievable. [ABSTRACT FROM AUTHOR]

We study distributed optimization in the presence of Byzantine adversaries, where both data and computation are distributed among $m$ worker machines, $t$ of which may be corrupt. The compromised nodes may collaboratively and arbitrarily deviate from their pre-specified programs, and a designated (master) node iteratively computes the model/parameter vector for generalized linear models. In this work, we primarily focus on two iterative algorithms: Proximal Gradient Descent (PGD) and Coordinate Descent (CD). Gradient descent (GD) is a special case of these algorithms. PGD is typically used in the data-parallel setting, where data is partitioned across different samples, whereas, CD is used in the model-parallelism setting, where data is partitioned across the parameter space. At the core of our solutions to both these algorithms is a method for Byzantine-resilient matrix-vector (MV) multiplication; and for that, we propose a method based on data encoding and error correction over real numbers to combat adversarial attacks. We can tolerate up to $t\leq \lfloor \frac {m-1}{2}\rfloor $ corrupt worker nodes, which is information-theoretically optimal. We give deterministic guarantees, and our method does not assume any probability distribution on the data. We develop a sparse encoding scheme which enables computationally efficient data encoding and decoding. We demonstrate a trade-off between the corruption threshold and the resource requirements (storage, computational, and communication complexity). As an example, for $t\leq \frac {m}{3}$ , our scheme incurs only a constant overhead on these resources, over that required by the plain distributed PGD/CD algorithms which provide no adversarial protection. To the best of our knowledge, ours is the first paper that connects MV multiplication with CD and designs a specific encoding matrix for MV multiplication whose structure we can leverage to make CD secure against adversarial attacks. Our encoding scheme extends efficiently to (i) the data streaming model, in which data samples come in an online fashion and are encoded as they arrive, and (ii) making stochastic gradient descent (SGD) Byzantine-resilient. In the end, we give experimental results to show the efficacy of our proposed schemes. [ABSTRACT FROM AUTHOR]

Limited magnitude asymmetric error model is well suited for flash memory. In this paper, we consider the construction of asymmetric codes correcting single error over $\mathbb {Z}_{2^{k}r}$ which is based on so called $B_{1}[{4}](2^{k}r)$ set. In fact, we reduce the construction of a maximal size $B_{1}[{4}](2^{k}r)$ set for $k\geq 3$ to the construction of a maximal size $B_{1}[{4}](2^{k-3}r)$ set. Finally, we give an explicit formula of a maximal size $B_{1}[{4}](4r)$ set and some lower bounds of a maximal size $B_{1}[{4}](2r)$ set. By computer searching up to $q\leq 106$ , we conjecture that those lower bounds are tight. [ABSTRACT FROM AUTHOR]

In this paper, we present a new method for explicitly constructing regular low-density parity-check (LDPC) codes based on \mathbb Sn( \mathbb {F}q) , the space of n\times n symmetric matrices over \mathbb {F}_{q} . Using this method, we obtain two classes of binary LDPC codes, {\mathcal{ C}}(n,q) and \mathcal C^T(n,q) , both of which have grith 8. Then, both the minimum distance and the stopping distance of each class are investigated. It is shown that the minimum distance and the stopping distance of \mathcal C^T(n,q) are both $2q$ . As for {\mathcal{ C}}(n,q) , we determine the minimum distance and the stopping distance for some special cases and obtain some lower bounds for other cases. [ABSTRACT FROM PUBLISHER]

A long-standing conjecture of Golomb and Welch, raised in 1970, states that there is no perfect r error correcting Lee code of length n for n\geq 3 and r>1 under the l_{p} metric, where 1\leq p<\infty . We show some nonexistence results of linear perfect lp codes for p=1 and 2\leq p<\infty , r=2^{1/p},3^{1/p} . We also give an algebraic construction of quasi-perfect l_{p}$ codes for p=1, r=2$ , and 2\leq p<\infty , r=2^{1/p} . [ABSTRACT FROM PUBLISHER]

Locally repairable codes (LRCs) are a class of codes designed for the local correction of erasures. They have received considerable attention in recent years due to their applications in distributed storage. Most existing results on LRCs do not explicitly take into consideration the field size $q$ , i.e., the size of the code alphabet. In particular, for the binary case, only a few results are known. In this paper, we present an upper bound on the minimum distance $d$ of linear LRCs with availability, based on the work of Cadambe and Mazumdar. The bound takes into account the code length $n$ , dimension $k$ , locality $r$ , availability $t$ , and field size $q$ . Then, we study the binary linear LRCs in three aspects. First, we focus on analyzing the locality of some classical codes, i.e., cyclic codes and Reed–Muller codes, and their modified versions, which are obtained by applying the operations of extend, shorten, expurgate, augment, and lengthen. Next, we construct LRCs using phantom parity-check symbols and multi-level tensor product structure, respectively. Compared with other previous constructions of binary LRCs with fixed locality or minimum distance, our construction is much more flexible in terms of code parameters, and gives various families of high-rate LRCs, some of which are shown to be optimal with respect to their minimum distance. Finally, the availability of LRCs is studied. We investigate the locality and availability properties of several classes of one-step majority-logic decodable codes, including cyclic simplex codes, cyclic difference-set codes, and 4-cycle free regular low-density parity-check codes. We also show the construction of a long LRC with availability from a short one-step majority-logic decodable code. [ABSTRACT FROM PUBLISHER]

Because of their capacity-approaching performance and their complexity/latency advantages, spatially-coupled (SC) codes are among the most attractive error-correcting codes for use in modern dense data storage systems. SC codes are constructed by partitioning an underlying block code and coupling the partitioned components. Here, we focus on circulant-based SC codes. Recently, the optimal overlap (OO), circulant power optimizer (CPO) approach was introduced to construct high performance SC codes for additive white Gaussian noise (AWGN) and Flash channels. The OO stage operates on the protograph of the SC code to derive the optimal partitioning that minimizes the number of graphical objects that undermine the performance of SC codes under iterative decoding. Then, the CPO optimizes the circulant powers to further reduce this number. Since the nature of detrimental objects in the graph of a code critically depends on the characteristics of the channel of interest, extending the OO-CPO approach to construct SC codes for channels with intrinsic memory is not a straightforward task. In this paper, we tackle one relevant extension; we construct high performance SC codes for practical 1-D magnetic recording channels, i.e., partial-response (PR) channels. Via combinatorial techniques, we carefully build and solve the optimization problem of the OO partitioning, focusing on the objects of interest in the case of PR channels. Then, we customize the CPO to further reduce the number of these objects in the graph of the code. SC codes designed using the proposed OO-CPO approach for PR channels outperform prior state-of-the-art SC codes by up to around 3 orders of magnitude in frame error rate (FER) and 1.1 dB in signal-to-noise ratio (SNR). More intriguingly, our SC codes outperform structured block codes of the same length and rate by up to around 1.8 orders of magnitude in FER and 0.4 dB in SNR. The performance advantage of SC codes designed using the devised OO-CPO approach over block codes of the same parameters is not only pronounced in the error floor region, but also in the waterfall region. [ABSTRACT FROM AUTHOR]

Instantaneous nonlocal quantum computation refers to a process in which spacelike separated parties simulate a nonlocal quantum operation on their joint systems through the consumption of pre-shared entanglement. To prevent a violation of causality, this simulation succeeds up to local errors that can only be corrected after the parties communicate classically with one another. However, this communication is non-interactive, and it involves just the broadcasting of local measurement outcomes. We refer to this operational paradigm as local operations and broadcast communication (LOBC) to distinguish it from the standard local operations and (interactive) classical communication (LOCC). In this paper, we show that an arbitrary two-qubit gate can be implemented by LOBC with $\epsilon $ -error using ${O}(\log (1/\epsilon))$ entangled bits (ebits). This offers an exponential improvement over the best known two-qubit protocols, whose ebit costs behave as ${O}(1/\epsilon)$. We also consider the family of binary controlled gates on dimensions ${d}_{A}\otimes {d}_{B}$. We find that any hermitian gate of this form can be implemented by LOBC using a single shared ebit. In sharp contrast, a lower bound of $\log {d}_{B}$ ebits is shown in the case of generic (i.e. non-hermitian) gates from this family, even when ${d}_{A}=2$. This demonstrates an unbounded gap between the entanglement costs of LOCC and LOBC gate implementation. Whereas previous lower bounds on the entanglement cost for instantaneous nonlocal computation restrict the minimum dimension of the needed entanglement, we bound its entanglement entropy. To our knowledge this is the first such lower bound of its kind. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that some results in the theory of rank-metric codes over finite fields can be extended to finite commutative principal ideal rings. More precisely, the rank metric is generalized and the rank-metric Singleton bound is established. The definition of Gabidulin codes is extended and it is shown that its properties are preserved. The theory of Gröbner bases is used to give the unique decoding, minimal list decoding, and error-erasure decoding algorithms of interleaved Gabidulin codes. These results are then applied in space-time codes and in random linear network coding as in the case of finite fields. Specifically, two existing encoding schemes of random linear network coding are combined to improve the error correction. [ABSTRACT FROM AUTHOR]

We construct error-correcting write-once memory (WOM) codes that guarantee correction of any specified number of errors in $q$ -level memories. The constructions use suitably designed short $q$ -ary WOM codes and concatenate them with outer error-correcting codes over different alphabets using suitably designed mappings. With a new storage-efficiency measure, we call EC-rate and show that for common error types the codes save redundancy and implementation complexity over straightforward concatenation. In addition to constructions for guaranteed error correction, we extend the error-correcting WOM scheme to binary multi-level coding for random errors, and toward soft-decision decoding provide an efficient way to extract reliability information without using higher-precision readout. [ABSTRACT FROM AUTHOR]

Most bounds on the size of codes hold for any code, whether linear or not. Notably, the Griesmer bound holds only in the linear case and so optimal linear codes are not necessarily optimal codes. In this paper, we identify code parameters $(q,d,k)$ , namely, field size, minimum distance, and combinatorial dimension, for which the Griesmer bound also holds in the (systematic) nonlinear case. Moreover, we show that the Griesmer bound does not necessarily hold for a systematic code by explicit construction of a family of optimal systematic binary codes. On the other hand, we are able to provide some versions of the Griesmer bound holding for all the systematic codes. [ABSTRACT FROM AUTHOR]

Random network coding recently attracts attention as a technique to disseminate information in a network. This paper considers a noncoherent multishot network, where the unknown and time-variant network is used several times. In order to create dependence between the different shots, particular convolutional codes in rank metric are used. These codes are so-called (partial) unit memory ((P)UM) codes, i.e., convolutional codes with memory one. First, distance measures for convolutional codes in rank metric are shown and two constructions of (P)UM codes in rank metric based on the generator matrices of maximum rank distance codes are presented. Second, an efficient error-erasure decoding algorithm for these codes is presented. Its guaranteed decoding radius is derived and its complexity is bounded. Finally, it is shown how to apply these codes for error correction in random linear and affine network coding. [ABSTRACT FROM AUTHOR]

Quantum synchronizable error-correcting codes are special quantum error-correcting codes that are designed to correct both the effect of quantum noise on qubits and misalignment in block synchronization. It is known that, in principle, such a code can be constructed through a combination of a classical linear code and its subcode if the two are both cyclic and dual-containing. However, finding such classical codes that lead to promising quantum synchronizable error-correcting codes is not a trivial task. In fact, although there are two families of classical codes that are proved to produce quantum synchronizable codes with good minimum distances and highest possible tolerance against misalignment, their code lengths have been restricted to primes and Mersenne numbers. In this paper, examining the incidence vectors of projective spaces over the finite fields of characteristic 2, we give quantum synchronizable codes from cyclic codes whose lengths are not primes or Mersenne numbers. These projective geometric codes achieve good performance in quantum error correction and possess the best possible ability to recover synchronization, thereby enriching the variety of good quantum synchronizable codes. We also extend the current knowledge of cyclic codes in classical coding theory by explicitly giving generator polynomials of the finite geometric codes and completely characterizing the minimum weight nonzero codewords. In addition to the codes based on projective spaces, we carry out a similar analysis on the well-known cyclic codes from Euclidean spaces that are known to be majority logic decodable and determine their exact minimum distances. [ABSTRACT FROM PUBLISHER]

We consider a problem in which a source is encoded into $N$ packets, an unknown number of which are subject to adversarial errors en route to the decoder. We seek code designs for which the decoder is guaranteed to be able to reproduce the source subject to a certain distortion constraint when there are no packets errors, subject to a less stringent distortion constraint when there is one error, and so on. Focusing on the special case of the erasure distortion measure, we introduce a code design based on the polytope codes of Kosut et al.. The resulting designs are also applied to a separate problem in distributed storage. [ABSTRACT FROM PUBLISHER]

Index coding is a source coding problem in which a broadcaster seeks to meet the different demands of several users, each of whom is assumed to have some prior information on the data held by the sender. A well-known application is satellite communications, as described in one of the earliest papers on the subject (Birk and Kol, 1998). It is readily seen that if the sender has knowledge of its clients’ requests and their side-information sets, then the number of packet transmissions required to satisfy all users’ demands can be greatly reduced if the data are encoded before sending. The collection of side-information indices as well as the indices of the requested data is described as an instance \mathcal I of the index coding with side-information (ICSI) problem. The encoding function is called the index code of $ \mathcal I$ , and the number of transmissions, resulting from the encoding is referred to as its length. The main ICSI problem is to determine the optimal length of an index code and instance $ \mathcal I$ . As this number is hard to compute, bounds approximating it are sought, as are algorithms to compute efficient index codes. These questions have been addressed by several authors (e.g., see Alon et al. 2008, Bar-Yossef et al. 2011, Blasiak et al. 2013), often taking a graph-theoretic approach. Two interesting generalizations of the problem that have appeared in the literature are the subject of this paper. The first of these is the case of index coding with coded side information (Dai et al. 2014), in which linear combinations of the source data are both requested by and held as users’ side-information. This generalization has applications, for example, to relay channels and necessitates algebraic rather than combinatorial methods. The second is the introduction of error-correction in the problem, in which the broadcast channel is subject to noise (Dau et al. 2013). In this paper, we characterize the optimal length of a scalar or vector linear index code with coded side information (ICCSI) over a finite field in terms of a generalized min-rank and give bounds on this number based on constructions of random codes for an arbitrary instance. We furthermore consider the length of an optimal $\delta $ -error correcting code for an instance of the ICCSI problem and obtain bounds analogous to those described in (Dau et al. 2013), both for the Hamming metric and for rank-metric errors. We describe decoding algorithms for both categories of errors. [ABSTRACT FROM AUTHOR]