Related papers: Three Combinatorial Perspectives on Minimal Codes
We study multi-budgeted variants of the classic minimum cut problem and graph separation problems that turned out to be important in parameterized complexity: Skew Multicut and Directed Feedback Arc Set. In our generalization, we assign…
Motivated by a question of Erd\H{o}s on blocking sets in a projective plane that intersect every line only a few times, several authors have used unions of algebraic curves to construct such sets in $\mathbb{P}^2(\mathbb{F}_q)$. In this…
The quadrature of cut elements is crucial for all Finite Element Methods that do not apply boundary-fitted meshes. It should be efficient, accurate, and robust. Various approaches balancing these requirements have been published, with some…
In distributed storage systems built using commodity hardware, it is necessary to have data redundancy in order to ensure system reliability. In such systems, it is also often desirable to be able to quickly repair storage nodes that fail.…
Nowadays there are several classes of constrained codes intended for different applications. The following two large classes can be distinguished. The first class contains codes with local constraints; for example, the source data must be…
Low-bit width neural networks have been extensively explored for deployment on edge devices to reduce computational resources. Existing approaches have focused on gradient-based optimization in a two-stage train-and-compress setting or as a…
Analytic combinatorics in several variables refers to a suite of tools that provide sharp asymptotic estimates for certain combinatorial quantities. In this paper, we apply these tools to determine the Gilbert--Varshamov lower bound on the…
We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of…
Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on…
A novel technique for construction of minimum storage regenerating (MSR) codes is presented. Based on this technique, three explicit constructions of MSR codes are given. The first two constructions provide access-optimal MSR codes, with…
The paper describes clustering problems from the combinatorial viewpoint. A brief systemic survey is presented including the following: (i) basic clustering problems (e.g., classification, clustering, sorting, clustering with an order over…
Using a finite geometric framework for studying the pentagon and heptagon codes we show that the concepts of quantum secret sharing and contextuality can be studied in a nice and unified manner. The basic idea is a careful study of the…
We study bounded width algebras which are minimal in the sense that every proper reduct does not have bounded width. We show that minimal bounded width algebras can be arranged into a pseudovariety with one basic ternary operation. We…
Following the approach by R. K\"otter and F. R. Kschischang, we study network codes as families of k-dimensional linear subspaces of a vector space F_q^n, q being a prime power and F_q the finite field with q elements. In particular,…
Linear constraints for a matrix polytope with no fractional vertex are investigated as intersecting research among permutation codes, rank modulations, and linear programming methods. By focusing the discussion to the block structure of…
Submodular set functions are undoubtedly among the most important building blocks of combinatorial optimization. Somewhat surprisingly, continuous counterparts of such functions have also appeared in an analytic line of research where they…
We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the…
In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound…
We adapt methods coming from additive combinatorics in groups to the study of linear span in associative unital algebras. In particular, we establish for these algebras analogues of Diderrich-Kneser's and Hamidoune's theorems on sumsets and…
We introduce two constructions of additive codes over finite fields. Both constructions start with a linear code over a field with $q$ elements and give additive codes over the field with $q^h$ elements whose minimum distance is…