Related papers: An Introduction to Algebraic Geometry codes
In this paper one presents a collection of results about the "bar codes" and "Jordan blocks" introduced by Burghelea-Day as "computer friendly" invariants of a tame angle-valued map and one relates these invariants to the Betti numbers,…
We demonstrate how one can see quantization of geometry, and quantum algebraic structure in supersymmetric gauge theory.
For Kummer extensions defined by $y^m = f (x)$, where $f (x)$ is a separable polynomial over the finite field $\mathbb{F}_q$, we compute the number of Weierstrass gaps at two totally ramified places. For many totally ramified places we give…
In this paper, we study residues of differential 2-forms on a smooth algebraic surface over an arbitrary field and give several statements about sums of residues. Afterwards, using these results we construct algebraic-geometric codes which…
The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of…
Power decoding is a partial decoding paradigm for arbitrary algebraic geometry codes for decoding beyond half the minimum distance, which usually returns the unique closest codeword, but in rare cases fails to return anything. The original…
We develop the basic theory of geometrically closed rings as a generalisation of algebraically closed fields, on the grounds of notions coming from positive model theory and affine algebraic geometry. For this purpose we consider several…
This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of…
Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work.…
The concept of viewing graph solvability has gained significant interest in the context of structure-from-motion. A viewing graph is a mathematical structure where nodes are associated to cameras and edges represent the epipolar geometry…
The minimum distance is one of the most important combinatorial characterizations of a code. The maximum likelihood decoding problem is one of the most important algorithmic problems of a code. While these problems are known to be hard for…
Fix a prime number $p$. We report on some recent developments in algebraic geometry (broadly construed) over $p$-adically complete commutative rings. These developments include foundational advances within the subject as well as external…
We give a brief introduction to computational algebraic number theory in OSCAR. Our main focus is on number fields, rings of integers and their invariants. After recalling some classical results and their constructive counterparts, we…
We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semi-algebraic sets. Aside from…
We describe a new link between the theory of topological modular forms and representations of vertex operator algebras obtained by certain lattices. The construction is motivated by the arithmetic Whitehead tower of the orthogonal groups.…
Fast encoding and decoding of codes have been always an important topic in code theory as well as complexity theory. Although encoding is easier than decoding in general, designing an encoding algorithm of codes of length $N$ with…
We describe various structures of algebraic nature on the space of continuous valuations on convex sets, their properties (like versions of Poincar\'e duality and hard Lefschetz theorem), and their relations and applications to integral…
In this paper we present an introduction to morphological calculus in which geometrical objects play the rule of generalised natural numbers.
This note is an invitation to the theory of geometric functions. The foundation techniques and some of the developments in the field are explained with the mindset that the audience is principally young researchers wishing to understand…
New families of unit memory as well as multi-memory convolutional codes are constructed algebraically in this paper. These convolutional codes are derived from the class of group character codes. The proposed codes have basic generator…