On the New Notion of the Set-Expectation for a Random Set of Events

The paper introduces new notion for the set-valued mean set of a random set. The means are defined as families of sets that minimize mean distances to the random set. The distances are determined by metrics in spaces of sets or by suitable generalizations. Some examples illustrate the use of the new definitions.mean random set, metrics in set space, mean distance, Aumann expectation, Frechet expectation, Hausdorff metric, random finite set, mean set, set-median, set-expectation

Constructions of Batch Codes via Finite Geometry

A primitive $k$-batch code encodes a string $x$ of length $n$ into string $y$ of length $N$, such that each multiset of $k$ symbols from $x$ has $k$ mutually disjoint recovering sets from $y$. We develop new explicit and random coding constructions of linear primitive batch codes based on finite geometry. In some parameter regimes, our proposed codes have lower redundancy than previously known batch codes.Comment: 7 pages, 1 figure, 1 tabl

On a games theory of random coalitions and on a coalition imputation

The main theorem of the games theory of random coalitions is reformulated in the random set language which generalizes the classical maximin theorem but unlike it defines a coalition imputation also. The theorem about maximin random coalitions has been introduced as a random set form of classical maximin theorem. This interpretation of the maximin theorem indicate the characteristic function of the game and its close connection with optimal random coalitions. So we can write the apparent natural formula of coalition imputation generalizing the strained formulas of imputation have been in the game theory till now. Those formulas of imputation we call the strained formulas because it is unknown from where the characteristic function of the game appears and because it is necessary to make additional suppositions about a type of distributions of random coalitions. The reformulated maximin theorem has both as its corollaries. The main outputs are two results of the games theory were united and the type of characteristic function of game defined by the game matrix was discovered.games theory, random coalition, coalition imputation

Symmetric Disjunctive List-Decoding Codes

A binary code is said to be a disjunctive list-decoding $s_L$-code (LD $s_L$-code), $s \ge 2$, $L \ge 1$, if the code is identified by the incidence matrix of a family of finite sets in which the union (or disjunctive sum) of any $s$ sets can cover not more than $L-1$ other sets of the family. In this paper, we consider a similar class of binary codes which are based on a {\em symmetric disjunctive sum} (SDS) of binary symbols. By definition, the symmetric disjunctive sum (SDS) takes values from the ternary alphabet $\{0, 1, *\}$, where the symbol~$*$ denotes "erasure". Namely: SDS is equal to $0$ ($1$) if all its binary symbols are equal to $0$ ($1$), otherwise SDS is equal to~$*$. List decoding codes for symmetric disjunctive sum are said to be {\em symmetric disjunctive list-decoding $s_L$-codes} (SLD $s_L$-codes). In the given paper, we remind some applications of SLD $s_L$-codes which motivate the concept of symmetric disjunctive sum. We refine the known relations between parameters of LD $s_L$-codes and SLD $s_L$-codes. For the ensemble of binary constant-weight codes we develop a random coding method to obtain lower bounds on the rate of these codes. Our lower bounds improve the known random coding bounds obtained up to now using the ensemble with independent symbols of codewords.Comment: 18 pages, 1 figure, 1 table, conference pape