We consider a generalization of the Pólya–Kasteleyn approach to counting the number of perfect matchings in a graph based on computing the symbolic Pfaffian of a directed adjacency matrix of the graph. Complexity of algorithms based on this approach is related to the complexity of the sign function of a perfect matching in generalized decision tree models. We obtain lower bounds on the complexity of

We review known results about codes that are able to protect multimedia content from illegal redistribution by coalitions of malicious users.

We consider an infinite directed graph with vertices numbered by integers \(\ldots,-2, -1,0,1,2,\ldots\strut\), where any pair of vertices \(j< k\) is connected by an edge \((j,k)\) that is directed from \(j\) to \(k\) and has a random weight \(v_{j,k}\in [-\infty,\infty)\). Here, \(\{v_{j,k},\: j< k\}\) is a family of independent and identically distributed random variables that take either finite

Motivated by applications in the security of cyber-physical systems, we pose the finite blocklength communication problem in the presence of a jammer as a zero-sum game between the encoder-decoder team and the jammer, by allowing the communicating team as well as the jammer only locally randomized strategies. The communicating team's problem is nonconvex under locally randomized codes, and hence, in

We prove that the maximum number of words in a code that corrects a fraction of \(1/4+\varepsilon\) of asymmetric errors in a Z-channel is \(\Theta(\varepsilon^{-3/2})\) as \(\varepsilon\to 0\).

In 1933, Borsuk stated a conjecture, which has become classical, that the minimum number of parts of smaller diameter into which an arbitrary set of diameter 1 in \(\mathbb{R}^n\) can be partitioned is \(n+1\). In 1993, this conjecture was disproved using sets of points with coordinates 0 and 1. Later, the second author obtained stronger counterexamples based on families of points with coordinates

We consider the problem of finding the minimum and maximum values of f-divergence for discrete probability distributions P and Q provided that one of these distributions and the value of their coupling are given. An explicit formula for the minimum value of the f-divergence under the above conditions is obtained, as well as a precise expression for its maximum value. This precise expression is not

We estimate the minimum distance of primary monomial affine variety codes defined from a hyperelliptic curve \({x^5} + x - {y^2}\) over \(\mathbb{F}_7\). To estimate the minimum distance of the codes, we apply symbolic computations implementing the techniques suggested by Geil and Özbudak. For some of these codes, we also obtain the symbol-pair distance. Furthermore, lower bounds on the generalized

We consider additive white Gaussian noise channels and discrete memoryless channels where the transmitter harvests energy from the environment. These can model wireless sensor networks as well as Internet of Things. By providing a unifying framework that works for any energy harvesting channel, we study these channels assuming an infinite energy buffer and provide the corresponding achievability and

We consider heterogeneous distributed storage systems (DSSs) having flexible reconstruction degree, where each node in the system has nonuniform repair bandwidth and nonuniform storage capacity. In particular, a data collector can reconstruct the file using some \(k\) nodes in the system and, for a node failure, the system can be repaired by some set of active nodes. Using min-cut bound, we investigate

We consider a generalized concatenated construction for error-correcting codes over the q-ary alphabet in the modulus metric L1 and Lee metric L. Resulting codes have arbitrary length, arbitrary distance (independently of the alphabet size), and can correct both independent errors and error bursts in both metrics. In particular, for any length 2m we construct codes over \(\mathbb{Z}_4\) with Lee distance

This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1134/S0032946020040092.

An underdetermined Bernoulli source generates symbols of a given underdetermined alphabet independently with some probabilities. To each underdetermined symbol there corresponds a set of basic (fully defined) symbols such that it can be substituted (specified) by any of them. An underdetermined source is characterized by its entropy, which is implicitly introduced as a minimum of a certain function

We complete the description of the procedure of topological expansion of a bipartite graph without parallel branches on the plane of changing the structure of cycles of length up to 10 inclusive. Based on previous papers, we have extended a set of theorems specifying transformation rules for cycles and paths when passing from a protograph to the Tanner graph. We propose a procedure for detecting the

It was shown very recently in [1] that there are no multimedia digital fingerprinting codes capable of fully recovering a coalition of malicious users under the general linear attack and adversarial noise. We show that such codes exist if the class of attacks is narrowed to the averaging attack. The arising mathematical problem is close to the problem of constructing signature codes for a noisy binary

In an NFV network, the availability of resource scheduling can be transformed to the existence of the fractional factor in the corresponding NFV network graph. Researching on the existence of special fractional factors in network structure can help to construct the NFV network with efficient application of resources. Let h: E(G) → [0, 1] be a function. We write \({d}_{G}^{h}(x)=\sum \limits_{e\ni x}h(e)\)

We consider narrow-sense BCH codes of length pm − 1 over \({{\mathbb{F}}}_{p}\), m ≥ 3. We prove that neither such a code with designed distance δ = 3 nor its extension for p ≥ 5 is generated by the set of its codewords of the minimum nonzero weight. We establish that extended BCH codes with designed distance δ = 3 for p ≥ 3 are generated by the set of codewords of weight 5, where basis vectors can

We study the asymptotic behavior of the independence number of a random subgraph of a certain (r, s)-distance graph. We provide upper and lower bounds for the critical edge survival probability under which a phase transition occurs, i.e., large new independent sets appear in the subgraph, which did not exist in the original graph.

A communication scenario is described involving a series of events triggered by a transmitter and observed by a receiver experiencing relativistic time dilation. The message selected by the transmitter is assumed to be encoded in the events’ timings and is required to be perfectly recovered by the receiver, regardless of the difference in clock rates in the two frames of reference. It is shown that

We analyze the shift-composition operation on discrete functions which occurs under homomorphisms of finite shift registers. We prove that for a prime p, in the class of all functions that are linear in the extreme variables, the notions of reducibility and p-linear reducibility coincide for p-linear functions. Furthermore, we show that a linear function irreducible in the class of all linear functions

Sphere packing bounds (SPBs)—with prefactors that are polynomial in the block length—are derived for codes on two families of memoryless channels using Augustin’s method: (possibly nonstationary) memoryless channels with (possibly multiple) additive cost constraints and stationary memoryless channels with convex constraints on the composition (i.e., empirical distribution, type) of the input codewords

We prove that any symmetric block design (v, k, λ) generates optimal ternary and quaternary constant-weight equidistant codes, whose parameters n, N, w, d, q are uniquely determined by the parameters of the block design. For one rather special case, we construct symbolwise uniform equidistant codes of the minimum length.

Let p be a prime number and s > 0 an integer. In this short note, we investigate one-point geometric Goppa codes associated with an elementary abelian p-extension of \({{\mathbb{F}}}_{{p}^{s}}(x)\). We determine their dimension and exact minimum distance in a few cases. These codes are a special case of weak Castle codes. We also list exact values of the second generalized Hamming weight of these codes

A graph G is called a fractional (g, f)-covered graph if for any e ∈ E(G), G admits a fractional (g, f)-factor covering e. A graph G is called a fractional (g, f, n)-critical covered graph if for any S ⊆ V(G) with ∣S∣ = n, G − S is a fractional (g, f)-covered graph. A fractional (g, f, n)-critical covered graph is said to be a fractional (a, b, n)-critical covered graph if g(x) = a and f(x) = b for

For a Gaussian two-armed bandit, which arises when batch data processing is analyzed, the minimax risk limiting behavior is investigated as the control horizon N grows infinitely. The minimax risk is searched for as the Bayesian one computed with respect to the worst-case prior distribution. We show that the highest requirements are imposed on the control in the domain of "close” distributions where

We consider the problem of adaptive estimation of a linear functional of an unknown multivariate vector from its observations against white Gaussian noise. As a family of estimators for the functional, we use those generated by projection estimators of the unknown vector, and the main problem is to select the best estimator in this family. The goal of the paper is to explain and mathematically justify

Contraction coefficients are distribution dependent constants that are used to sharpen standard data processing inequalities for f-divergences (or relative f-entropies) and produce so-called “strong” data processing inequalities. For any bivariate joint distribution, i.e., any probability vector and stochastic matrix pair, it is known that contraction coefficients for f-divergences are upper bounded

We consider a hypothesis testing problem where a part of data cannot be observed. Our helper observes the missed data and can send us a limited amount of information about them. What kind of this limited information will allow us to make the best statistical inference? In particular, what is the minimum information sufficient to obtain the same results as if we directly observed all the data? We derive

For cycles of length 8 in a Tanner graph, we propose an identification procedure based on the analysis of paths in a protograph. We formulate and prove a number of theorems that introduce identification rules for cycles and restrict the number of subgraphs to be analyzed. To distinguish between them, we propose a number of parameters that uniquely determine the group of analyzed paths in the protograph

A Steiner triple system (STS) contains a transversal subdesign TD(3, w) if its point set has three pairwise disjoint subsets A, B, C of size w and w2 blocks of the STS intersect with each of A, B, C (those w2 blocks form a TD(3,w)). We prove several structural properties of Steiner triple systems of order 3w + 3 that contain one or more transversal subdesigns TD(3, w). Using exhaustive search, we find

We describe the construction of a piecewise polynomial generator over a Galois ring and prove a transitivity criterion for it. We give an estimate for the discrepancy of the output sequences of such a generator. We show that the obtained estimate is asymptotically equivalent to known estimates for special cases of a piecewise polynomial generator, and in some cases it is asymptotically sharper.

We consider q-ary block codes with exactly two distances: d and d + 1. Several constructions of such codes are given. In the linear case, we show that all codes can be obtained by a simple modification of linear equidistant codes. Upper bounds for the maximum cardinality of such codes are derived. Tables of lower and upper bounds for small q and n are presented.

Orthogonal arrays play an important role in statistics and experimental design. Like other combinatorial constructions, the most important and studied problems are questions about their existence and classification. An essential step to solving such problems is determination of Hamming distance distributions of an orthogonal array with given parameters. In this paper we propose an algorithm for computing

We consider the problem of finding maximum values of f-divergences Df(P ∥ Q) of discrete probability distributions P and Q with values on a finite set under the condition that the variation distance V(P, Q) between them and one of the distributions P or Q are given. We obtain exact expressions for such maxima of f-divergences, which in a number of cases allow to obtain both explicit formulas and upper

We study the maximum remaining service time in M(2)∣G2∣∞ fork -join queueing systems where an incoming task forks on arrival for service into two subtasks, each of them being served in one of two infinite-sever subsystems. The following cases for the arrival rate are considered: (1) time-independent, (2) given by a function of time, (3) given by a stochastic process. As examples of service time distributions

We establish an extended large deviation principle for processes with independent and stationary increments on the half-line under the Cramer moment condition in the space of functions of bounded variation without discontinuities of the second kind equipped with the Borovkov metric.

This paper concerns the folklore statement that “entropy is a lower bound for compression.” More precisely, we derive from the entropy theorem a simple proof of a pointwise inequality first stated by Ornstein and Shields and which is the almost-sure version of an average inequality first stated by Khinchin in 1953. We further give an elementary proof of the original Khinchin inequality, which can be

We consider the moving element search problem with the minimum total cardinality of tests. As a search space, we consider the set of integer points of a segment of length n. We prove that the total test cardinality of an asymptotically optimal adaptive strategy is \(n + 2\sqrt n \).

By a (Latin) unitrade of order k, we call a subset of vertices of the Hamming graph H(n, k) that intersects any maximal clique at either 0 or 2 vertices. A bitrade is a bipartite unitrade, i.e., a unitrade that can be split into two independent subsets. We study the cardinality spectrum of bitrades in the Hamming graph H(n, k) with k = 3 (ternary hypercube) and the growth of the number of such bitrades

We refine a lower bound on the spectrum of a binary code. We give a simple derivation of the known bound on the undetected error probability of a binary code.

For any channel, the existence of a unique Augustin mean is established for any positive order and probability mass function on the input set. The Augustin mean is shown to be the unique fixed point of an operator defined in terms of the order and the input distribution. The Augustin information is shown to be continuously differentiable in the order. For any channel and convex constraint set with

We derive an analog of the Frankl-Wilson theorem on independence numbers of some distance graphs. The obtained results are applied to the problem of the chromatic number of a space ℝn with a forbidden equilateral triangle and to the problem of chromatic numbers of distance graphs with large girth.

In an earlier paper we developed a unified approach to the extendability problem for arcs in PG(k - 1, q) and, equivalently, for linear codes over finite fields. We defined a special class of arcs called (t mod q)-arcs and proved that the extendabilty of a given arc depends on the structure of a special dual arc, which turns out to be a (t mod q)-arc. In this paper, we investigate the general structure

We present a family of easily computable upper bounds for the Holevo (information) quantity of an ensemble of quantum states depending on a reference state (as a free parameter). These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that an appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many

We consider the problem of finding the maximum values of divergences D(P‖Q) and D(Q‖P) for probability distributions P and Q ranging in the finite set \(\mathcal{N}=\left\{1,\;2,...,n\right\}\) provided that both the variation distance V (P,Q) between them and either the probability distribution Q or (in the case of D(P‖Q)) only the value of the minimal component qmin of the probability distribution

We slightly improve the superconcentrator construction by Alon and Capalbo.

We consider alphabetic coding of superwords. We establish an unambiguity coding criterion for the cases of finite and infinite codes. We prove that in the case of an infinite code the ambiguity detection problem is m-complete in the ∃1∀0 class of Kleene’s analytical hierarchy.

A quarter century ago Chor, Fiat, and Naor proposed mathematical models for revealing a source of illegal redistribution of digital content (tracing traitors) in the broadcast encryption framework, including the following two combinatorial models: nonbinary IPP codes, based on an (n, n)-threshold secret sharing scheme, and IPP set systems, based on the general (w, n)-threshold secret sharing scheme

Codes with the identifiable “parent” property appeared as one of solutions for the broadcast encryption problem. We propose a new, more general model of such codes, give an overview of known results, and formulate some unsolved problems.

Cyclic codes as a subclass of linear codes have practical applications in communication systems, consumer electronics, and data storage systems due to their efficient encoding and decoding algorithms. The objective of this paper is to construct some cyclic codes by the sequence approach. More precisely, we determine the dimension and the generator polynomials of three classes of q-ary cyclic codes

Abstract—We correct mistakes in the formulations of Theorem 19 and Proposition 17 of the original article, published in vol. 55, no. 1, 1–45.

The impact of diversity on reliable communication over arbitrarily varying channels (AVC) is investigated as follows. First, the concept of an identical state-constrained jammer is motivated. Second, it is proved that symmetrizability of binary symmetric AVCs (AVBSC) caused by identical state-constrained jamming is circumvented when communication takes place over at least three orthogonal channels

We introduce a new wide class of error-correcting codes, called non-split toric codes. These codes are a natural generalization of toric codes where non-split algebraic tori are taken instead of usual (i.e., split) ones. The main advantage of the new codes is their cyclicity; hence, they can possibly be decoded quite fast. Many classical codes, such as (doubly-extended) Reed-Solomon and (projective)

The operation of Minkowski addition of geometric figures has a discrete analog, addition of subsets of a Boolean cube viewed as a vector space over the two-element field. Subsets of the Boolean cube (or multivariable Boolean functions) form a monoid with respect to this operation. This monoid is of interest in classical discrete analysis as well as in a number of problems related to information theory

We use Hamilton equations to identify most likely scenarios of long queues being formed in ergodic Jackson networks. Since the associated Hamiltonians are discontinuous and piecewise Lipschitz, one has to invoke methods of nonsmooth analysis. Time reversal of the Hamilton equations yields fluid equations for the dual network. Accordingly, the optimal trajectories are time reversals of the fluid trajectories

We prove equivalence of using the modulus metric and Euclidean metric in solving the soft decoding problem for a memoryless discrete channel with binary input and Q-ary output. For such a channel, we give an example of a construction of binary codes correcting t binary errors in the Hamming metric. The constructed codes correct errors at the output of a demodulator with Q quantization errors as (t

This paper considers a multimessage network where each node may send a message to any other node in the network. Under the discrete memoryless model, we prove the strong converse theorem for any network whose cut-set bound is tight, i.e., achievable. Our result implies that for any fixed rate vector that resides outside the capacity region, the average error probability of any sequence of length-n

An exact expression for the probability of inversion of a large spin is established in the form of an asymptotic expansion in the series of Bessel functions with orders belonging to an arithmetic progression. Based on the new asymptotic expansion, a formula for the inversion time of the spin is derived.

We consider the problem of determining extreme values of the Rényi entropy for a discrete random variable provided that the value of the α-coupling for this random variable and another one with a given probability distribution is fixed.

This work is a survey on completely regular codes. Known properties, relations with other combinatorial structures, and construction methods are considered. The existence problem is also discussed, and known results for some particular cases are established. In addition, we present several new results on completely regular codes with covering radius ρ = 2 and on extended completely regular codes.