Geometric construction of some families of two-class and three-class association schemes and codes from nondegenerate and degenerate Hermitian varieties

AbstractTaking a nondegenerate Hermitian variety as a projective set in a projective plane PG(2,s2), Mesner (1967) derived a two-class association scheme on the points of the affine space of dimension 3, for which the projective plane is the plane at infinity.We generalize his construction in two ways. We show how his construction works both for nondegenerate and degenerate Hermitian varieties in any dimension.We consider a projective space of dimension N, partitioned into an affine space of dimension N and a hyperplane H of dimension N − 1 at infinity.The points of the hyperplane are next partitioned into 2 or 3 subsets. A pair of points a,b of the affine space is defined to belong to class i if the line ab meets the subset i of H.In the first case, the two subsets of the hyperplane are a nondegenerate Hermitian variety and its complement. In this case, we show that the classification of pairs of affine points defines a family of two-class association schemes. This family of association schemes has the same set of parameters as those derived as restrictions of the Hamming association schemes to two-weight codes defined as linear spans of coordinate vectors of points on a nondegenerate Hermitian variety in a projective space of dimension N − 1. The relations of these codes to orthogonal arrays and difference sets are described in [5,6].In the second case, the three subsets are the singular point of the variety, the regular points of the variety and the complement of the variety defined by a Hermitian form of rank N − 1. This leads to a family of three-class association schemes on the points of the affine space. A geometric construction is first given for the case N = 3.Using a general algebraic method pointed out by the referee, we have also derived the three-class association scheme for general N

Intersection sets, three-character multisets and associated codes

In this article we construct new minimal intersection sets in ${\mathrm{AG}}(r,q^2)$ sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in ${\mathrm{PG}}(r,q^2)$ with $r$ even and we also compute their weight distribution.Comment: 17 Pages; revised and corrected result

Construction of Rational Surfaces Yielding Good Codes

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 "\`a la Weil" of Aubry and Perret together with the bound of Homma and Kim for plane curves. The parameters of several codes from rational surfaces are computed. Among them, the codes defined by the evaluation of forms of degree 3 on an elliptic quadric are studied. As far as we know, such codes have never been treated before. Two other rational surfaces are studied and very good codes are found on them. In particular, a [57,12,34] code over $\mathbf{F}_7$ and a [91,18,53] code over $\mathbf{F}_9$ are discovered, these codes beat the best known codes up to now.Comment: 20 pages, 7 figure

Sums of residues on algebraic surfaces and application to coding theory

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 are an extension to surfaces of the well-known differential codes on curves. We also study some properties of these codes and extend to them some known properties for codes on curves.Comment: 31 page

Quaternion Algebras

This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282