1,903 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology
We study smooth higher symmetry groups and moduli -stacks of generic
higher geometric structures on manifolds. Symmetries are automorphisms which
cover non-trivial diffeomorphisms of the base manifold. We construct the smooth
higher symmetry group of any geometric structure on and show that this
completely classifies, via a universal property, equivariant structures on the
higher geometry. We construct moduli stacks of higher geometric data as
-categorical quotients by the action of the higher symmetries, extract
information about the homotopy types of these moduli -stacks, and prove
a helpful sufficient criterion for when two such higher moduli stacks are
equivalent.
In the second part of the paper we study higher -connections.
First, we observe that higher connections come organised into higher groupoids,
which further carry affine actions by Baez-Crans-type higher vector spaces. We
compute a presentation of the higher gauge actions for -gerbes with
-connection, comment on the relation to higher-form symmetries, and present
a new String group model. We construct smooth moduli -stacks of higher
Maxwell and Einstein-Maxwell solutions, correcting previous such considerations
in the literature, and compute the homotopy groups of several moduli
-stacks of higher - connections. Finally, we show that a
discrepancy between two approaches to the differential geometry of NSNS
supergravity (via generalised and higher geometry, respectively) vanishes at
the level of moduli -stacks of NSNS supergravity solutions.Comment: 102 pages; comments welcom
Several families of ternary negacyclic codes and their duals
Constacyclic codes contain cyclic codes as a subclass and have nice algebraic
structures. Constacyclic codes have theoretical importance, as they are
connected to a number of areas of mathematics and outperform cyclic codes in
several aspects. Negacyclic codes are a subclass of constacyclic codes and are
distance-optimal in many cases. However, compared with the extensive study of
cyclic codes, negacyclic codes are much less studied. In this paper, several
families of ternary negacyclic codes and their duals are constructed and
analysed. These families of negacyclic codes and their duals contain
distance-optimal codes and have very good parameters in general
Minimal PD-sets for codes associated with the graphs Qm2, m even
Please read abstract in the article.The National Research Foundation of South Africahttp://link.springer.com/journal/2002021-12-08hj2021Mathematics and Applied Mathematic
On Linear Equivalence, Canonical Forms, and Digital Signatures
The LESS signature scheme, introduced in 2020, represents a fresh research direction to obtain practical code-based signatures. LESS is based on the linear equivalence problem for codes, and the scheme is entirely described using matrices, which define both the codes, and the maps between them. It makes sense then, that the performance of the scheme depends on how efficiently such objects can be represented.
In this work, we investigate canonical forms for matrices, and how these can be used to obtain very compact signatures. We present a new notion of equivalence for codes, and prove that it reduces to linear equivalence; this means there is no security loss when applying canonical forms to LESS. Additionally, we flesh out a potential application of canonical forms to cryptanalysis, and conclude that this does not improve on existing attacks, for the regime of interest.
Finally, we analyze the impact of our technique, showing that it yields a drastic reduction in signature size when compared to the LESS submission, resulting in the smallest sizes for code-based signature schemes based on zero-knowledge
An imperceptible connection between the Clebsch--Gordan coefficients of and the Terwilliger algebras of Grassmann graphs
The Clebsch--Gordan coefficients of are expressible in
terms of Hahn polynomials. The phenomenon can be explained by an algebra
homomorphism from the universal Hahn algebra into
. Let denote a finite
set and denote the power set of . It is generally known that
supports a -module. Fix an element
. By the linear isomorphism given by for all , this induces a
-module structure on . Pulling back via the algebra homomorphism , the -module forms an -module.
The -module enfolds the Terwilliger algebra
of a Johnson graph. This result connects these two seemingly irrelevant topics:
The Clebsch--Gordan coefficients of and the Terwilliger
algebras of Johnson graphs. Unfortunately some steps break down in the
-analog case. By making detours, the imperceptible connection between the
Clebsch--Gordan coefficients of and the Terwilliger
algebras of Grassmann graphs is successfully disclosed in this paper.Comment: 65 page
-Additive Hadamard Codes
The -additive codes are subgroups of
, and can be seen as linear codes over
when , -additive or -additive
codes when or , respectively, or
-additive codes when . A
-linear Hadamard code is a Hadamard code
which is the Gray map image of a
-additive code. In this paper, we
generalize some known results for -linear Hadamard
codes to -linear Hadamard codes with
, , and . First, we give a
recursive construction of -additive
Hadamard codes of type with
, , and . Then, we show that in general the
-linear, -linear and
-linear Hadamard codes are not included in the family
of -linear Hadamard codes with , , and . Actually, we point out that
none of these nonlinear -linear Hadamard
codes of length is equivalent to a
-linear Hadamard code of any other type,
a -linear Hadamard code, or a
-linear Hadamard code, with , of the same length
Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats
International audienceAs data is a predominant resource in applications, Riemannian geometry is a natural framework to model and unify complex nonlinear sources of data.However, the development of computational tools from the basic theory of Riemannian geometry is laborious.The work presented here forms one of the main contributions to the open-source project geomstats, that consists in a Python package providing efficient implementations of the concepts of Riemannian geometry and geometric statistics, both for mathematicians and for applied scientists for whom most of the difficulties are hidden under high-level functions. The goal of this monograph is two-fold. First, we aim at giving a self-contained exposition of the basic concepts of Riemannian geometry, providing illustrations and examples at each step and adopting a computational point of view. The second goal is to demonstrate how these concepts are implemented in Geomstats, explaining the choices that were made and the conventions chosen. The general concepts are exposed and specific examples are detailed along the text.The culmination of this implementation is to be able to perform statistics and machine learning on manifolds, with as few lines of codes as in the wide-spread machine learning tool scikit-learn. We exemplify this with an introduction to geometric statistics
Equivariant theory for codes and lattices I
In this paper, we present a generalization of Hayden's theorem [7, Theorem
4.2] for -codes over finite Frobenius rings. A lattice theoretical form of
this generalization is also given. Moreover, Astumi's MacWilliams identity [1,
Theorem 1] is generalized in several ways for different weight enumerators of
-codes over finite Frobenius rings. Furthermore, we provide the Jacobi
analogue of Astumi's MacWilliams identity for -codes over finite Frobenius
rings. Finally, we study the relation between -codes and its corresponding
-lattices.Comment: 22 page
- …