Related papers: Coding Theory and Algebraic Combinatorics
The following open problems, which concern a fundamental limit on coding properties of quantum codes with realistic physical constraints, are analyzed and partially answered here: (a) the upper bound on code distances of quantum…
The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of…
Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…
Due to the increased complexity of software development projects more and more systems are described by models. The sheer size makes it impractical to describe these systems by a single model. Instead many models are developed that provide…
The major challenge in designing a discriminative learning algorithm for predicting structured data is to address the computational issues arising from the exponential size of the output space. Existing algorithms make different assumptions…
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep…
The authors describe their approach to teaching a course on finite fields and combinatorial applications, including block designs and error-correcting codes, using a hybrid of lectures and active learning. Under the discussed classroom…
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…
This paper, following (Dymetman:1998), presents an approach to grammar description and processing based on the geometry of cancellation diagrams, a concept which plays a central role in combinatorial group theory (Lyndon-Schuppe:1977). The…
In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and…
Starting from the (apparently) elementary problem of deciding how many different topological spaces can be obtained by gluing together in pairs the faces of an octahedron, we will describe the central role played by hyperbolic geometry…
This paper provides a survey of spherical designs and their applications, with a particular emphasis on the perspective of ``numerical analysis''. A set \(X_N\) of \(N\) points on the unit sphere \(\mathbb{S}^d\) is called a…
We review the main results of the theory of rank-metric codes, with emphasis on their combinatorial properties. We study their duality theory and MacWilliams identities, comparing in particular rank-metric codes in vector and matrix…
Harnessing the potential computational advantage of quantum computers for machine learning tasks relies on the uploading of classical data onto quantum computers through what are commonly referred to as quantum encodings. The choice of such…
In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
This paper is primarily intended as an introduction for the mathematically inclined to some of the rich algebraic combinatorics arising in for instance CFT. It is essentially self-contained, apart from some of the background motivation and…
We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of…
Shannon's seminal 1948 work gave rise to two distinct areas of research: information theory and mathematical coding theory. While information theory has had a strong influence on theoretical neuroscience, ideas from mathematical coding…
We first find the combinatorial degree of any map $f:V\to F$ where $F$ is a finite field and $V$ is a finite-dimensional vector space over $F$. We then simplify and generalize a certain construction due to Chein and Goodaire that was used…