Search

The topological interference management (TIM) problem refers to the study of the $K$ -user partially connected interference networks with no channel state information at the transmitters (CSIT), except for the knowledge of network topology. In this paper, we study the TIM problem with confidential messages (TIM-CM), where message confidentiality must be satisfied in addition to reliability constraints. In particular, each transmitted message must be decodable at its intended receiver and remain confidential at the remaining $(K-1)$ receivers. Our main contribution is to present a comprehensive set of results for the TIM-CM problem by studying the symmetric secure degrees of freedom (SDoF). To this end, we first characterize necessary and sufficient conditions for feasibility of positive symmetric SDoF for any arbitrary topology. We next present two achievable schemes for the TIM-CM problem: For the first scheme, we use the concept of secure partition and, for the second one, we use the concept of secure independent sets. We also present outer bounds on symmetric SDoF for any arbitrary network topology. Using these bounds, we characterize the optimal symmetric SDoF of all $K=2$ -user and $K=3$ -user network topologies. [ABSTRACT FROM AUTHOR]

In high-dimensional sparse regression, would increasing the signal-to-noise ratio while fixing the sparsity level always lead to better model selection? For high-dimensional sparse regression problems, surprisingly, in this paper we answer this question in the negative in the regime of linear sparsity for the Lasso method, relying on a new concept we term effect size heterogeneity. Roughly speaking, a regression coefficient vector has high effect size heterogeneity if its nonzero entries have significantly different magnitudes. From the viewpoint of this new measure, we prove that the false and true positive rates achieve the optimal trade-off uniformly along the Lasso path when this measure is maximal in a certain sense, and the worst trade-off is achieved when it is minimal in the sense that all nonzero effect sizes are roughly equal. Moreover, we demonstrate that the first false selection occurs much earlier when effect size heterogeneity is minimal than when it is maximal. The underlying cause of these two phenomena is, metaphorically speaking, the “competition” among variables with effect sizes of the same magnitude in entering the model. Taken together, our findings suggest that effect size heterogeneity shall serve as an important complementary measure to the sparsity of regression coefficients in the analysis of high-dimensional regression problems. Our proofs use techniques from approximate message passing theory as well as a novel technique for estimating the rank of the first false variable. [ABSTRACT FROM AUTHOR]

This paper describes new, simple, recursive methods of construction for orientable sequences, i.e. periodic binary sequences in which any $n$ -tuple occurs at most once in a period in either direction. As has been previously described, such sequences have potential applications in automatic position-location systems, where the sequence is encoded onto a surface and a reader needs only examine $n$ consecutive encoded bits to determine its location and orientation on the surface. The only previously described method of construction (due to Dai et al.) is somewhat complex, whereas the new techniques are simple to both describe and implement. The methods of construction cover both the standard ‘infinite periodic’ case, and also the aperiodic, finite sequence, case. Both the new methods build on the Lempel homomorphism, first introduced as a means of recursively generating de Bruijn sequences. [ABSTRACT FROM AUTHOR]

The tight lower bound on the redundancy of optimal prefix-free codes in terms of a given symbol probability (not necessarily the largest or the smallest one) has been derived in the literature. The first goal of this paper is to derive the tight lower bounds in terms of $j$ ($j>1$) known symbol probabilities of the source using some properties of Kullback-Leibler distance. Since fix-free codes are special prefix-free codes (for which no codeword is a suffix of the other codewords), it is clear that all lower bounds on the redundancy of optimal prefix-free codes are also valid for fix-free ones. Accordingly, the second question of the paper is on the tightness of the derived lower bounds for optimal fix-free codes. It is proven that these lower bounds are tight for optimal fix-free codes if $j\leq 3$ , and are not tight for $j\geq 4$. Also, it is shown that the tight lower bound in terms of the probability of the most likely symbol is the same for optimal prefix-free and optimal fix-free codes. [ABSTRACT FROM AUTHOR]

We investigate the list decodability of symbol-pair codes in this paper. First, we show that the list decodability of every symbol-pair code does not exceed the Gilbert–Varshamov bound. On the other hand, we are able to prove that with high probability, a random symbol-pair code can be list decoded up to the Gilbert–Varshamov bound. Our second result of this paper is to derive the Johnson-type bound, i.e., a lower bound on list decoding radius in terms of minimum distance. Finally, we present a list decoding algorithm of Reed–Solomon codes beyond the Johnson-type bound in the pair metric. 1 A symbol-pair code is referred to a code in the pair metric. [ABSTRACT FROM AUTHOR]

In a paper by Guruswami et al., it was shown that the length $n$ of a $q$ -ary linear locally recoverable code with distance $d \geqslant 5$ is upper bounded by $O(dq^{3})$. Thus, it is a challenging problem to construct $q$ -ary locally recoverable codes with distance $d \geqslant 5$ and length approaching the upper bound. The same paper also gave an algorithmic construction of $q$ -ary locally recoverable codes with locality $r$ and length $n=\Omega _{r}(q^{2})$ for $d=5$ and 6, where $\Omega _{r}$ means that the implicit constant depends on locality $r$. In this paper, we present an explicit construction of $q$ -ary locally recoverable codes of distance $d= 5$ and 6 via binary constant weight codes. It turns out that 1) our construction is simpler and more explicit and 2) the length of our codes is greater than previously known. [ABSTRACT FROM AUTHOR]

The zero capacity region for hexagonal $(d,k)$ run-length constraints is known for many, but not all, $d$ and $k$. The pairs $(d,k)$ for which it has been unproven whether the capacity is zero or positive consist of: (i) $k=d+2$ when $d\ge 2$ ; (ii) $k=d+3$ when $d \ge 1$ ; (iii) $k=d+4$ when either $d=4$ or $d$ is odd and $d \ge 3$ ; and (iv) $k=d+5$ when $d=4$. Here, we prove that the capacity is zero for all of case (i), and for case (ii) whenever $d \ge 7$. The method used in this paper is to reduce an infinite search space of valid labelings to a finite set of configurations that we exhaustively examine using backtracking. In Part II of this two-part series, we use automated procedures to prove that the capacity is zero in case (i) when $2 \le d \le 9$ , in case (ii) when $3 \le d \le 11$ , and in case (iii) when $d \in \{ 4,5,7,9 \}$ , and that the capacity is positive in case (ii) when $d \in \{1,2\}$ , in case (iii) when $d = 3$ , and in case (iv). Thus, the only remaining unknown cases are now when $k=d+4$ , for any odd $d \ge 11$. [ABSTRACT FROM AUTHOR]

We study the stability of low-density parity-check codes under blockwise or bitwise maximum a posteriori decoding, where transmission takes place over a binary-input memoryless output-symmetric channel. Our study stems from the consideration of constructing universal capacity-achieving codes under low-complexity decoding algorithms, where universality refers to the fact that we are considering a family of channels with equal capacity. Consider, e.g., the right-regular sequence by Shokrollahi and the heavy-tail Poisson sequence by Luby et al. Both sequences are provably capacity-achieving under belief propagation decoding when transmission takes place over the binary erasure channel. In this paper we show that many existing capacity-achieving sequences of low-density parity-check codes are not universal under belief propagation decoding. We reveal that the key to showing this non-universality result is determined by the stability of the underlying codes. More concretely, for an ordered and complete channel family and a sequence of low-density parity-check code ensembles, we determine a stability threshold associated with them, which gives rise to a sufficient condition under which the sequence is not universal under belief propagation decoding. Moreover, we show that the same stability threshold applies to blockwise or bitwise maximum a posteriori decoding as well. We demonstrate how the stability threshold can determine an upper bound on the corresponding blockwise or bitwise maximum a posteriori threshold, revealing the operational significance of the stability threshold. [ABSTRACT FROM AUTHOR]

A complete characterization for the restricted isometry constant (RIC) bounds on $\delta _{{{ tk}}}$ for all $ {t}>0$ is an important problem on recovery of sparse signals with prior support information via weighted $\ell _{{p}}$ -minimization ($0 < {p} \leqslant 1$). In this paper, new bounds on the restricted isometry constants $\delta _{{{ tk}}}$ ($0 < {t} < \frac {4}{3}{d}$), where $d$ is a key constant determined by prior support information, are established to guarantee the sparse signal recovery via the weighted $\ell _{{p}}$ minimization in both noiseless and noisy settings. This result fills a vacancy on $\delta _{{{ tk}}}$ with $0 < {t} < \frac {4}{3}{d}$ , compared with previous works on $\delta _{{{ tk}}}$ (${t} \geqslant \frac {4}3{d}$). We show that, when the accuracy of prior support estimate is at least 50%, the new recovery condition in terms of $\delta _{{{ tk}}}$ ($0 < {t} < \frac {4}{3}{d}$) via weighted $\ell _{1}$ minimization is weaker than the condition required by classical $\ell _{1}$ minimization without weighting. Our weighted $\ell _{1}$ minimization gives better recovery error bounds in noisy setting. Similarly, the new recovery condition in terms of $\delta _{{{ tk}}}$ ($0 < {t} < \frac {4}{3}{d}$) is extended to weighted $\ell _{{p}}$ ($0 < {p} < 1$) minimization, and it is also weaker than the condition obtained by standard non-convex $\ell _{{p}}$ ($0 < {p} < 1$) minimization without weighting. Numerical illustrations are provided to demonstrate our new theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we provide two complementary results for cyclic codes of composite length. First, we give a general construction of cyclic codes of length nr from cyclic codes of length n and a lower bound on the minimum distance. Numerical data show that many cyclic codes of composite length with the best parameters can be obtained in this way. Second, in the other direction, we show that for cyclic codes of length n with a primitive n-th root of unity as a non-zero, the minimum distance is roughly bounded by the square-free part of n. This means that we shall not expect good cyclic codes of length n in general if, for example, the length n is divisible by a high power of a prime. [ABSTRACT FROM AUTHOR]

The Demirci-Selçuk’s meet-in-the-middle attack is one of the most important methods among all the cryptanalytic vectors, which gives the best result against the round-reduced AES with respect to the rounds, and tradeoffs between data, time and memory. While we have already built provable security models against the differential cryptanalysis, linear cryptanalysis cryptanalysis, impossible differential and zero-correlation linear cryptanalysis, the provable security against the meet-in-the-middle attack is missing. In this paper, we propose the subset representation of function based on which we could give an algorithm to compute the exact number of parameters of the Demirci-Selçuk’s distinguisher given the input and output, respectively. Experiments show that this algorithm can be more efficient than the automatical tool presented by Shi et al. at Asiacrypt 2018. We further extract a formula based on this algorithm and show an upper bound for the length of the Demirci-Selçuk’s distinguisher of an iterative SPN cipher. We prove that for an SPN block cipher whose block size equals the key size, an effective Demirci-Selçuk-type meet-in-the-middle distinguisher covers at most twice the maximum of the primitive indexes of the linear layer and its inverse. As a result, we show that the known length of the Demirci-Selçuk’s distinguisher of the AES-128 cannot be improved unless the details of the S-boxes are exploited, which demonstrates that the AES has a provable security against the Demirci-Selçuk’s meet-in-the-middle attack. [ABSTRACT FROM AUTHOR]

Consider a channel with a given input alphabet size and input distribution. Our aim is to degrade or upgrade it to a channel with at most L output letters. Channel Q is degraded with respect to a channel W if Q can be obtained from W by processing the output of W. Upgrading is the inverse relation. This paper contains four main results. The first result, from which the paper title is derived, deals with the so called “greedy–merge” algorithm. We derive an upper bound on the reduction in mutual information between input and output, as a function of L. This upper bound is within a constant factor of an algorithm-independent lower bound. Thus, we establish that greedy–merge is optimal in the power-law sense (i.e. the power of L). The other main results deal with upgrading. The second result shows that a certain sequence of channels, which was previously shown to be “hard” for degrading, displays the same hardness in the context of upgrading. That is, suppose we are given such a channel and a corresponding input distribution. If we upgrade (degrade) to a new channel with L output letters, we incur an increase (decrease) in mutual information between input and output. We show that a previously derived bound on the decrease in mutual information for the degrading case is also a lower bound on the increase for the upgrading case. The third result is an efficient algorithm for optimal upgrading, in the binary-input case. That is, we are given a channel and an input distribution. We must find an upgraded channel with L output letters, for which the increase in mutual information is minimal. We give a simple characterization of such a channel, which implies an efficient algorithm. The fourth result is an analog of the first result for the upgrading case when the input is binary. That is, we first present a sub-optimal algorithm for the setting considered in the third result. The main advantage of the sub-optimal algorithm is that it is amenable to analysis. We carry out the analysis and show that the increase incurred in mutual information is within a constant factor of the lower bound derived in the second result. [ABSTRACT FROM AUTHOR]

Tietäväinen’s upper and lower bounds assert that for block-length- $n$ linear codes with dual distance $d$ , the covering radius $R$ is at most $({n}/{2})-(({1}/{2})-o(1))\sqrt {dn}$ and typically at least $({n}/{2})-\Theta (({dn\log {({n}/{d})}})^{1/2})$. The gap between those bounds on $R -({n}/{2})$ is a $\Theta (({\log {({n}/{d})}})^{1/2})$ factor related to the gap between the worst covering radius given $d$ and the sphere-covering bound. Our focus in this paper is on the case when $d = o(n)$ , i.e., when the code size is subexponential and the gap is $w(1)$. We show that up to a constant, the gap can be eliminated by relaxing the covering requirement to allow for missing $o(1)$ fraction of points. Namely, if the dual distance $d = o(n)$ , then for sufficiently large $d$ , almost all points can be covered with radius $R\leq ({n}/{2})-\Theta (({dn\log {({n}/{d})}})^{1/2})$. Compared with random linear codes, our bound on $R-({n}/{2})$ is asymptotically tight up to a factor less than 3. We give applications to dual-BCH codes. The proof builds on the author’s previous work on the weight distribution of cosets of linear codes, which we simplify in this paper and extend from codes to probability distributions on $\{0,1\}^{n}$ , thus enabling the extension of the earlier result to $(d-1)$ -wise independent distributions. [ABSTRACT FROM AUTHOR]

A primitive $k$ -batch code encodes a string $\boldsymbol {x}$ of length $n$ into a string $\boldsymbol {y}$ of length $N$ , such that each multiset of $k$ symbols from $\boldsymbol {x}$ has $k$ mutually disjoint recovering sets from $\boldsymbol {y}$. In this paper, we discuss new constructions of binary primitive batch codes. First, we develop novel explicit and random coding constructions of linear primitive batch codes based on finite geometries. Second, a new explicit coding construction of binary primitive batch codes based on bivariate lifted multiplicity codes is provided. For any $k=n^ \varepsilon $ with $\varepsilon \in (0,0.47)\setminus \{1/5,1/4\}$ , our proposed codes have a better trade-off between the redundancy and the parameters $k,n$ than previously known batch codes. [ABSTRACT FROM AUTHOR]

Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called rank metric codes with given ranks (GRMCs), to generalize the parallel construction in [Xu and Chen, IEEE Trans. Inf. Theory, 64 (2018), 6315–6319] and the classic multilevel construction. A Singleton-like upper bound and a lower bound for GRMCs derived from Gabidulin codes are given. Via GRMCs, two effective constructions for CDCs are presented by combining the parallel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given. The ratio between the new lower bound and the known upper bound for $(4\delta,2\delta,2\delta)_{q}$ -CDCs is calculated. It is greater than 0.99926 for any prime power $q$ and any $\delta \geq 3$. [ABSTRACT FROM AUTHOR]

A nonlinear feedback shift register (NFSR) of $n$ -stage is called irreducible if, the family of output sequences of any NFSR of stage less than $n$ is not included in that of the NFSR. Tian and Qi in this paper [IEEE-IT, 2013(6),4006–4012] gave a lower bound on the density of irreducible NFSRs. In this paper, we improve their lower bound and also give an upper bound on the density of irreducible NFSRs. Moreover, the gap between our upper and lower bounds is less than 0.04. [ABSTRACT FROM PUBLISHER]

Using some recent results on asymptotically equivalent sequences of matrices, we present in this paper, a new derivation of the capacity formula given by Brandenburg and Wyner for a discrete-time Gaussian multiple-input-multiple-output channel with memory. In this paper, we tackle not only the case considered by them, where the number of channel inputs and the number of channel outputs are the same, but also when both numbers are different. [ABSTRACT FROM AUTHOR]

Locally repairable codes have been investigated extensively in recent years due to practical applications in distributed and cloud storage systems. However, there are few asymptotic constructions of locally repairable codes in the literature. In this paper, we provide a new explicit asymptotic construction of locally repairable codes over arbitrary finite fields from local expansions of functions at a rational place. This construction gives a Tsfasman-Vladut-Zink type bound for locally repairable codes. Its main advantage is that there are no constraints on both locality and alphabet size. Furthermore, we show that the Gilbert-Varshamov type bound of locally repairable codes over non-prime finite fields can be improved for sufficiently large alphabet sizes. [ABSTRACT FROM AUTHOR]

Locally repairable codes with more than one recovering set are demanded in the application to distributed storage. For each failure node (or disk), it is desired to have as many recovering sets as possible. In this paper, we make use of automorphisms of rational function fields to construct locally repairable codes with multiple recovering sets. Although we focus on two recovering sets, our construction can be easily generalized to the case of multiple recovering sets. In particular, we obtain a class of locally repairable codes with minimum distance only 1 less than the upper bound. [ABSTRACT FROM AUTHOR]

Locally recoverable codes are very important due to their applications in distributed storage systems. In this paper, by using cyclic codes, we construct two classes of optimal q-ary cyclic $(\text {r}, \delta _{1})$ locally recoverable codes with parameters $[2(\text {q}+1), 2(\text {q}+1)-2\delta _{1}, \delta _{1}+2]_{\text {q}}$ , where $q$ is an odd prime power and $\text {r}+\delta _{1}-1=\text {q}+1$ , and $(\text {r}, \delta _{2})$ locally recoverable codes with parameters $[2(\text {q}+1), 2(\text {q}+1)-4\delta _{2}+2, \delta _{2}+2]_{q}$ , where $\text {q}\geq 7$ is an odd prime power, $\text {q}\equiv 3~ (mod~ 4)$ and $r+\delta _{2}-1=\frac {\text {q}+1}{2}$. Compared with the known cyclic and constacyclic $(\text {r}, \delta)$ locally recoverable codes, our construction yields new optimal cyclic $(\text {r}, \delta)$ locally recoverable codes. [ABSTRACT FROM AUTHOR]

This paper studies the noncausal relay channel, also known as the relay channel with unlimited look ahead, introduced by by El Gamal, Hassanpour, and Mammen. Unlike the standard relay channel model, where the relay encodes its signal based on the previous received output symbols, the relay in the noncausal relay channel encodes its signal as a function of the entire received sequence. In the existing coding schemes, the relay uses this noncausal information solely to recover the transmitted message or part of it and then cooperates with the sender to communicate this message to the receiver. However, it is shown in this paper that by applying the Gelfand–Pinsker coding scheme, the relay can take further advantage of the noncausally available information and achieve rates strictly higher than those of the existing coding schemes. This paper also provides a new upper bound on the capacity of the noncausal relay channel that strictly improves upon the existing cutset bound. These new lower and upper bounds on the capacity coincide for the class of degraded noncausal relay channels and establish the capacity for this class. [ABSTRACT FROM AUTHOR]

This paper studies codes that correct a burst of deletions or insertions. Namely, a code will be called a $b$ -burst-deletion/insertion-correcting code if it can correct a burst of deletions/insertions of any $b$ consecutive bits. While the lower bound on the redundancy of such codes was shown by Levenshtein to be asymptotically $\log (n)+b-1$ , the redundancy of the best code construction by Cheng et al. is $b(\log (n/b+1))$ . In this paper, we close on this gap and provide codes with redundancy at most $\log (n) + (b-1)\log (\log (n)) +b -\log (b)$ . We first show that the models of insertions and deletions are equivalent and thus it is enough to study codes correcting a burst of deletions. We then derive a non-asymptotic upper bound on the size of $b$ -burst-deletion-correcting codes and extend the burst deletion model to two more cases: 1) a deletion burst of at most $b$ consecutive bits and 2) a deletion burst of size at most $b$ (not necessarily consecutive). We extend our code construction for the first case and study the second case for $b=3,4$ . [ABSTRACT FROM PUBLISHER]

This paper studies the problem of information symbol availability in codes: we refer to a systematic code as code with $(r, t)$ -availability if every information (systematic) symbol can be reconstructed from $t$ disjoint groups of other code symbols, each of the sizes at most $r$ . This paper shows that it is possible to construct codes that can support a scaling number of parallel reads while keeping the rate to be an arbitrarily high constant. It further shows that this is possible with the minimum Hamming distance arbitrarily close to the Singleton bound. This paper also presents a bound demonstrating a tradeoff between rate, minimum Hamming distance, and availability parameters. Our codes match the aforementioned bound, and their constructions rely on certain combinatorial structures. Resolvable designs provide one way to realize these required combinatorial structures. The two constructions presented in this paper require field sizes, which are linear and exponential in the code length, respectively. From a practical standpoint, our codes are relevant for distributed storage applications involving hot data, i.e., the information, which is frequently accessed by multiple processes in parallel. [ABSTRACT FROM AUTHOR]

New capacity upper bounds are presented for the discrete-time Poisson channel with no dark current and an average-power constraint. These bounds are a consequence of techniques developed for the seemingly unrelated problem of upper bounding the capacity of binary deletion and repetition channels. Previously, the best known capacity upper bound in the regime where the average-power constraint does not approach zero was due to Martinez (JOSA B, 2007), which is re-derived as a special case of the framework developed in this paper. Furthermore, this framework is carefully instantiated in order to obtain a closed-form bound that improves the result of Martinez everywhere. Finally, capacity-achieving distributions for the discrete-time Poisson channel are studied under an average-power constraint and/or a peak-power constraint and arbitrary dark current. In particular, it is shown that the support of the capacity-achieving distribution under an average-power constraint must only be countably infinite. This settles a conjecture of Shamai (IEE Proceedings I, 1990) in the affirmative. Previously, it was only known that the support must be an unbounded set. [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]

When a node in a distributed storage system fails, it needs to be promptly repaired to maintain system integrity. While typical erasure codes can provide a significant storage advantage over replication, they suffer from poor repair efficiency. Locally repairable codes (LRCs) tackle this issue by reducing the number of nodes participating in the repair process (locality), at the cost of reduced minimum distance. In this paper, we study the tradeoff between locality and minimum distance of LRCs with local codes that have arbitrary distance requirements. Unlike existing methods, where both the locality and the local distance requirements imposed on every node are identical, we allow the requirements to vary arbitrarily from node to node. Such a property can be an advantage for distributed storage systems with non-homogeneous characteristics. We present Singleton-type distance upper bounds and also provide an optimal code construction with respect to these bounds. In addition, the feasible rate region is characterized by dimension upper bounds that do not depend on the 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]

The main problem of constant-dimension subspace coding is to determine the maximal possible size ${\mathbf{A}}_{q}(n,d,k)$ of a set of $k$ -dimensional subspaces in ${\mathbf{F}}_{q}^{n}$ such that the subspace distance satisfies $d(U,V) \geq d$ for any two different subspaces $U$ and $V$ in this set. In this paper, we give a direct construction of constant-dimension subspace codes from two parallel versions of maximum rank-distance codes. The problem about the sizes of our constructed constant-dimension subspace codes is transformed into finding a suitable sufficient condition to restrict number of the roots of $L_{1}(L_{2}(x))-x$ where $L_{1}$ and $L_{2}$ are $q$ -polynomials over the extension field ${\mathbf{F}}_{q^{n}}$. New lower bounds for ${\mathbf{A}}_{q}(4k,2k,2k)$ , ${\mathbf{A}}_{q}(4k+2,2k,2k+1)$ , and ${\mathbf{A}}_{q}(4k+2,2(k-1),2k+1)$ are presented. Many new constant-dimension subspace codes better than previously best known codes with small parameters are constructed. [ABSTRACT FROM AUTHOR]

List decoding of insertions and deletions in the Levenshtein metric is considered. The Levenshtein distance between two sequences is the minimum number of insertions and deletions needed to turn one of the sequences into the other. In this paper, a Johnson-like upper bound on the maximum list size when list decoding in the Levenshtein metric is derived. This bound depends only on the length and minimum Levenshtein distance of the code, the length of the received word, and the alphabet size. It shows that polynomial-time list decoding beyond half the Levenshtein distance is possible for many parameters. Further, we also prove a lower bound on list decoding of deletions with the well-known binary Varshamov–Tenengolts codes, which shows that the maximum list size grows exponentially with the number of deletions. Finally, an efficient list decoding algorithm for two insertions/deletions with VT codes is given. This decoder can be modified to a polynomial-time list decoder of any constant number of insertions/deletions. [ABSTRACT FROM AUTHOR]

In this paper, we study the minimum/stopping distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers \(q\) and \(m\) , where \(q\) is an odd prime and \(m \leq q\) . In the literature, the minimum/stopping distance of these codes (denoted by \(d(q,m)\) and \(h(q,m)\) , respectively) has been thoroughly studied for \(m \leq 5\) . Both exact results, for small values of \(q\) and \(m\) , and general (i.e., independent of \(q\) ) bounds have been established. For \(m=6\) , the best known minimum distance upper bound, derived by Mittelholzer, is \(d(q,6) \leq 32\) . In this paper, we derive an improved upper bound of \(d(q,6) \leq 20\) and a new upper bound \(d(q,7) \leq 24\) by using the concept of a template support matrix of a codeword/stopping set. The bounds are tight with high probability in the sense that we have not been able to find codewords of strictly lower weight for several values of \(q\) using a minimum distance probabilistic algorithm. Finally, we provide new specific minimum/stopping distance results for \(m \leq 7\) and low-to-moderate values of \(q \leq 79\) . [ABSTRACT FROM AUTHOR]

For non-binary codes the Elias bound is a good upper bound for the asymptotic information rate at low-relative minimum distance, whereas the Plotkin bound is better at high-relative minimum distance. In this paper, we obtain a hybrid of these bounds, which improves both. This in turn is based on the anticode bound, which is a hybrid of the Hamming and Singleton bounds and improves both bounds. The question of convexity of the asymptotic rate function is an important open question. We conjecture a much weaker form of the convexity, and we show that our bounds follow immediately if we assume the conjecture. [ABSTRACT FROM AUTHOR]

The Stinson–Wei traceability scheme (known as traceability scheme) was proposed for broadcast encryption as a generalization of the Chor–Fiat–Naor traceability scheme (known as traceability code). Cover-free family was introduced by Kautz and Singleton in the context of binary superimposed code. In this paper, we find a new relationship between a traceability scheme and a cover-free family, which strengthens the anti-collusion strength from t to t^{2} , i.e., a t$ -traceability scheme is a t^{2} -cover-free family. Based on this interesting discovery, we derive new upper bounds for traceability schemes. By using combinatorial structures, we construct several infinite families of optimal traceability schemes, which attain our new upper bounds. We also provide a constructive lower bound for traceability schemes, the size of which has the same order of magnitude as our general upper bound. Meanwhile, we consider parent-identifying set systems, an anti-collusion key-distributing scheme requiring weaker conditions than traceability scheme but stronger conditions than cover-free family. A new upper bound is also given for parent-identifying set systems. [ABSTRACT FROM PUBLISHER]

Wachter-Zeh showed that every cover metric code can be list decoded up to the Johnson-like bound. Furthermore, it was shown that the efficient list decoding of cover metric codes up to the Johnson-like bound can be performed. From the work of Wachter-Zeh, one natural question is whether the Johnson-like bound can be improved. In this paper, we give a confirmative answer to this question by showing that the cover metric codes can be list decoded up to the Singleton bound. Our contributions consist of three parts. First, we prove that the list decodability of cover metric codes does not exceed the Singleton bound. Second, we show that, with high probability, a random cover metric code can be list decoded up to the Singleton bound, which is better than the Johnson-like bound. Third, by applying the existing decoding algorithms for Hamming metric and rank metric codes, we present explicit constructions of cover metric codes that can be efficiently list decoded up to the Singleton bound. [ABSTRACT FROM AUTHOR]

In pliable index coding, we consider a server with m messages and n clients, where each client has as side information a subset of the messages. We seek to minimize the number of broadcast transmissions, so that each client can recover any one unknown message she does not already have. Previous work has shown that the pliable index coding problem is NP-hard and requires at most \mathcal O(\log ^2(n)) broadcast transmissions, which indicates exponential savings over the conventional index coding that requires in the worst case \mathcal O(n) transmissions. In this paper, building on a decoding criterion that we propose, we first design a deterministic polynomial-time algorithm that can realize the exponential benefits, by achieving, in the worst case, a performance upper bounded by \mathcal O(\log ^2(n)) broadcast transmissions. We extend our algorithm to the t -requests case, where each client requires t unknown messages that she does not have, and show that our algorithm requires at most \mathcal O(t\log (n)+\log ^2(n)) broadcast transmissions. We construct lower bound instances that require at least \Omega (\log (n)) transmissions for linear pliable index coding and at least $\Omega (t+\log (n))$ transmissions for the t$ -requests case, indicating that both our upper and lower bounds are polynomials of \log (n)$ and differ within a factor of . We provide a probabilistic analysis over random instances and show that the required number of transmissions is almost surely $\Theta (\log (n))$ , as compared with the $\Theta (n/\log (n))$ for index coding. In addition, we show that these upper and lower bounds also hold for vector pliable index coding in the worst case instances and the random graph instances, implying that vector coding does not provide benefits in terms of these bounds. Our numerical experiments show that our algorithm outperforms existing algorithms for pliable index coding by up to 50% less transmissions. [ABSTRACT FROM AUTHOR]

In recent literature, a family of optimal linear locally recoverable codes (LRC codes) that attain the maximum possible distance (given code length, cardinality, and locality) is presented. The key ingredient for constructing such optimal linear LRC codes is the so-called $r$ -good polynomials, where r$ is equal to the locality of the LRC code. However, given a prime p$ , known constructions of r exist only for some special integers r$ , and the problem of constructing optimal LRC codes over small field for any given locality is still open. In this paper, by using function composition, we present two general methods of designing good polynomials, which lead to three new constructions of r$ -good polynomials. Such polynomials bring new constructions of optimal LRC codes. In particular, our constructed polynomials as well as the power functions yield optimal (n,k,r) for all positive integers $r$ as localities, where $q$ is near the code length $n$ . [ABSTRACT FROM AUTHOR]

The Johnson-type upper bound on the maximum size of a code of length n , distance d=2w-1 , and constant composition \overline w is \lfloor \dfrac \vphantom R^.nw1\rfloor , where w is the total weight and w_{1} is the largest component of {\overline {w}} . Recently, Chee et al. proved that this upper bound can be achieved for all constant-composition codes of sufficiently large lengths. Let Nccc({\overline {w}}) be the smallest such length. The determination of Nccc({\overline {w}}) is trivial for binary codes. This paper provides a lower bound on Nccc({\overline {w}}) , which is shown to be tight for all ternary and quaternary codes by giving new combinatorial constructions. Consequently, by the refining method, we determine the values of Nccc({\overline {w}}) , for all $q$ -ary constant-composition codes, provided that 3w_{1}\geq w with finite possible exceptions. [ABSTRACT FROM AUTHOR]

Frameproof codes are used to fingerprint digital data. They can prevent copyrighted materials from unauthorized use. In this paper, we study upper and lower bounds for $w$ -frameproof codes of length N$ over an alphabet of size q$ . The upper bound is based on a combinatorial approach and the lower bound is based on a probabilistic construction. Both bounds can improve one of the previous results when q$ is small compared with w$ , say cq\leq w$ for some constant c\leq q$ . Furthermore, we pay special attention to binary frameproof codes. We show a binary w . [ABSTRACT FROM PUBLISHER]

Caching is a promising solution to satisfy the ever-increasing demands for the multi-media traffics. In caching networks, coded caching is a recently proposed technique that achieves significant performance gains over the uncoded caching schemes. However, to implement the coded caching schemes, each file has to be split into F packets, which usually increases exponentially with the number of users K . Thus, designing caching schemes that decrease the order of F is meaningful for practical implementations. In this paper, by reviewing the Ali-Niesen caching scheme, the placement delivery array (PDA) design problem is first formulated to characterize the placement issue and the delivery issue with a single array. Moreover, we show that, through designing appropriate PDA, new centralized coded caching schemes can be discovered. Second, it is shown that the Ali-Niesen scheme corresponds to a special class of PDA, which realizes the best coding gain with the least F . Third, we present a new construction of PDA for the centralized coded caching system, wherein the cache size M at each user (identical cache size is assumed at all users) and the number of files N ( q is an integer, such that q\geq 2 ). The new construction can decrease the required F of Ali-Niesen scheme to O(e^{K\cdot ({M}/{N})\ln \!\,({N}/\!{M})}) or O(e^{K\cdot (1-({M}/\!{N}))\ln \!\,(N/(N-M))})$ , respectively, while the coding gain loss is only 1. [ABSTRACT FROM AUTHOR]

Suppose Y^n is obtained by observing a uniform Bernoulli random vector X^n through a binary symmetric channel. Courtade and Kumar asked how large the mutual information between Y^n and a Boolean function \mathsf b(X^n) could be, and conjectured that the maximum is attained by a dictator function. An equivalent formulation of this conjecture is that dictator minimizes the prediction cost in a sequential prediction of Y^n under logarithmic loss, given \mathsf b(X^n) . In this paper, we study the question of minimizing the sequential prediction cost under a different (proper) loss function—the quadratic loss. In the noiseless case, we show that majority asymptotically minimizes this prediction cost among all Boolean functions. We further show that for weak noise, majority is better than a dictator, and that for a strong noise dictator outperforms majority. We conjecture that for quadratic loss, there is no single sequence of Boolean functions that is simultaneously (asymptotically) optimal at all noise levels. [ABSTRACT FROM PUBLISHER]

Multiply constant-weight codes (MCWCs) have been recently studied to improve the reliability of certain physically unclonable function response. In this paper, we give combinatorial constructions for the MCWCs, which yield several new infinite families of optimal MCWCs. Furthermore, we demonstrate that the Johnson-type upper bounds of the MCWCs are asymptotically tight for fixed Hamming weights and distances. Finally, we provide bounds and constructions of the 2-D MCWCs. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of the universal coding of natural numbers. A new numeration system is introduced, which is based on variable- $r$ meta-Fibonacci sequences and it is a generalization of the Zeckendorf numeration system. This new numeration system is used to construct binary, prefix-free, uniquely decodable universal codes called meta-Fibonacci codes. The main advantage of these codes is that they are parametrized by a sequence of numbers, the sequence o. By controlling the growth of the values of this sequence, we can control the length of the code word. This means that we can provide a general framework for building efficient universal coders for natural numbers. Such framework is applied to the upper bounds of the code word length defined by Leung-Yan-Cheong and Cover (1978), Levenshtein (1968), and Ahlswede (1997). There is no other code meeting these bounds. In each case, we build meta-Fibonacci codes and demonstrate that the upper bound of their code word length is satisfied up to an additive constant, thereby solving these open problems. The framework may be applied to other upper bounds that satisfy Kraft inequality. [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]

The focus of this paper is on the public communication required for generating a maximal-rate secret key (SK) within the multiterminal source model of Csiszár and Narayan. Building on the prior work of Tyagi for the two-terminal scenario, we derive a lower bound on the communication complexity, R \text {SK} , defined to be the minimum rate of public communication needed to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by R \text {CO} , is an upper bound on R \text {SK} . For the class of pairwise independent network (PIN) models defined on uniform hypergraphs, we show that a certain Type \mathcal S condition, which is verifiable in polynomial time, guarantees that our lower bound on R \text {SK} meets the R \text {CO} upper bound. Thus, the PIN models satisfying our condition are R \text {SK} -maximal, indicating that the upper bound R \text {SK} \le R \text {CO} holds with equality. This allows us to explicitly evaluate R \text {SK} for such PIN models. We also give several examples of PIN models that satisfy our Type \mathcal S condition. Finally, we prove that for an arbitrary multiterminal source model, a stricter version of our Type \mathcal S condition implies that communication from all terminals (omnivocality) is needed for establishing an SK of maximum rate. For three-terminal source models, the converse is also true: omnivocality is needed for generating a maximal-rate SK only if the strict Type \mathcal S condition is satisfied. However, for the source models with four or more terminals, counterexamples exist showing that the converse does not hold in general. [ABSTRACT FROM PUBLISHER]

In this paper, the difference between the redundancy of the optimal asymmetric/symmetric fix-free code, and that of the optimal prefix-free code is considered as the penalty of benefiting from the desired properties of fix-free codes. This penalty is studied from different perspectives. In particular, it is shown that the average penalty of asymmetric fix-free codes is less than 0.21 bit per symbol. Moreover, it is proved that when the source alphabet size is sufficiently large, for almost all sources, the penalty is less than or equal to 0.182 bit per symbol. Regarding symmetric fix-free codes, it is shown that the average penalty tends to infinity as the source alphabet size increases. [ABSTRACT FROM AUTHOR]

The problem of detecting a single anomalous process among a finite number $M$ of processes is considered. At each time, a subset of the processes can be observed, and the observations from each chosen process follow two different distributions, depending on whether the process is normal or abnormal. The objective is a sequential search strategy that minimizes the expected detection time subject to an error probability constraint. This problem can be considered as a special case of active hypothesis testing first considered by Chernoff where a randomized strategy, referred to as the Chernoff test, was proposed and shown to be asymptotically (as the error probability approaches zero) optimal. For the special case considered in this paper, we show that a simple deterministic test achieves asymptotic optimality and offers better performance in the finite regime. We further extend the problem to the case where multiple anomalous processes are present. In particular, we examine the case where only an upper bound on the number of anomalous processes is known. [ABSTRACT FROM PUBLISHER]

The study of Armstrong codes is motivated by the problem of understanding complexities of dependences in relational database systems, where attributes have bounded domains. A $(q,k,n)$ -Armstrong code is a $q$ -ary code of length $n$ with minimum Hamming distance $n-k+1$ , and for any set of $k-1$ coordinates, there exist two codewords that agree exactly there. Let $f(q,k)$ be the maximum $n$ for which such a code exists. In this paper, $f(q,3)=3q-1$ is determined for all $q\geq 5$ with three possible exceptions. This disproves a conjecture of Sali. Furthermore, we introduce generalized Armstrong codes for branching, or $(s,t)$ -dependences, construct several classes of optimal Armstrong codes, and establish lower bounds for the maximum length $n$ in this more general setting. [ABSTRACT FROM AUTHOR]

Write-once memory (WOM) is a storage medium with memory elements, called cells, which can take on \(q\) levels. Each cell is initially in level 0 and can only increase its level. A \(t\) -write WOM code is a coding scheme, which allows one to store \(t\) messages to the WOM such that on consecutive writes every cell’s level does not decrease. The sum-rate of the WOM code, which is the ratio between the total amount of information written in the \(t\) writes and number of memory cells, is bounded by \(\log (t+1)\) . Our main contribution in this paper is a construction of binary three-write WOM codes with sum-rate approaching 1.885 for sufficiently large number of cells, whereas the upper bound is 2. This improves upon a recent construction of sum-rate 1.809. A key ingredient in our construction is a recent capacity achieving construction of two-write WOM codes, which uses the so-called Wozencraft ensemble of linear codes. In our construction, we encode information in the first and second write in a way that leaves a large number (roughly half) of the cells nonprogrammed. This allows us to use the above two-write construction in order to invoke a third write to the memory. We also give specific constructions of nonbinary two-write WOM codes and multiple writes, which give better sum-rate than the currently best known ones. In the construction of these codes, we build upon previous nonbinary constructions and show how tools such symbols relabeling can help in achieving high sum-rates. [ABSTRACT FROM PUBLISHER]

Multimedia fingerprinting is an effective technique to trace the sources of pirate copies of copyrighted multimedia contents. Separable codes were introduced to detect colluders taking part in the averaging attack, which is the most feasible approach to perform a collusion attack. In this paper, we provide some improved bounds for the size of separable codes. Using a combinatorial technique named grouping coordinates, we have greatly reduced the upper bound. On the other hand, we also provide some lower bounds for the size of separable codes by both probabilistic and deterministic construction methods. In particular, \( \overline {2}\) -separable codes with asymptotically optimal rate are obtained by the deletion method. [ABSTRACT FROM AUTHOR]