A translation of Problemy Peredachi Informatsii

Volume 57, Number 2, April–June, 2021
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Minimax Theorems for Finite Blocklength Lossy Joint Source-Channel Coding over an Arbitrarily Varying Channel
A. S. Vora and A. A. Kulkarni
pp. 99–128

Abstract—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 general, a minimax theorem need not hold for this game. However, we show that approximate minimax theorems hold in the sense that the minimax and maximin values of the game approach each other asymptotically. In particular, for rates strictly below a critical threshold, both the minimax and maximin values approach zero, and for rates strictly above it, they both approach unity. We then show a second-order minimax theorem, i.e., for rates exactly approaching the threshold along a specific scaling, the minimax and maximin values approach the same constant value, that is neither zero nor one. Critical to these results is our derivation of finite blocklength bounds on the minimax and maximin values of the game and our derivation of second-order dispersion-based bounds.


Coding in a Z-Channel in Case of Many Errors
V. S. Lebedev and N. A. Polyanskii
pp. 129–135

Abstract—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 $\varepsilon\to 0$.


Bounds on Borsuk Numbers in Distance Graphs of a Special Type
A. V. Berdnikov and A. M. Raigorodskii
pp. 136–142

Abstract—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 $-1$, $0$, and $1$. We establish new lower bounds for Borsuk numbers in families of this type.


Counting the Number of Perfect Matchings, and Generalized Decision Trees
M. N. Vyalyi
pp. 143–160

Abstract—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 the sign of a perfect matching in terms of the deterministic communication complexity of the XOR sign function of a matching. These bounds demonstrate limitations of the symbolic Pfaffian method for both the general case and the case of sparse graphs.


Limit Theorems for the Maximal Path Weight in a Directed Graph on the Line with Random Weights of Edges
T. Konstantopoulos, A. V. Logachov, A. A. Mogulskii, and S. G. Foss
pp. 161–177

Abstract—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 values (of any sign) or the value $-\infty$. A path in the graph is a sequence of connected edges $(j_0,j_1),(j_1,j_2),\ldots,(j_{m-1},j_m)$ (where $j_0<j_1<\ldots <j_m$), and its weight is the sum $\sum\limits_{s=1}^m v_{j_{s-1},j_s}\ge -\infty$ of the weights of the edges. Let $w_{0,n}$ be the maximal weight of all paths from $0$ to $n$. Assuming that $\operatorname{\mathbf P}(v_{0,1}>0)>0$, that the conditional distribution of $\operatorname{\mathbf P}(v_{0,1}\in\cdot\,\,|\, v_{0,1}>0)$ is nondegenerate, and that $\operatorname{\mathbf E}\exp (Cv_{0,1})<\infty$ for some $C=\mathrm{const}>0$, we study the asymptotic behavior of random sequence $w_{0,n}$ as $n\to\infty$. In the domain of the normal and moderately large deviations we obtain a local limit theorem when the distribution of random variables $v_{i,j}$ is arithmetic and an integro-local limit theorem if this distribution is non-lattice.


Separable Collusion-Secure Multimedia Codes
E. E. Egorova and G. A. Kabatiansky
pp. 178–198

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