401 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
On Chevalley group schemes over function fields: quotients of the Bruhat-Tits building by -arithmetic subgroups
Let be a reductive Chevalley group scheme (defined over
). Let be a smooth, projective, geometrically
integral curve over a field . Let be a closed point on
. Let be the ring of functions that are regular outside
. The fraction field of has a discrete valuation
associated to . In this
work, we study the action of the group of -points of
on the Bruhat-Tits building
in order to describe the
structure of the orbit space . We obtain
that this orbit space is the ``gluing'' of a closed connected CW-complex with
some sector chambers. The latter are parametrized by a set depending on the
Picard group of and on the rank of
. Moreover, we observe that any rational sector face whose tip is a
special vertex contains a subsector face that embeds into this orbit space. We
deduce, from this description, a writing of as a free product
with amalgamation. We also obtain a counting of the -conjugacy classes
of maximal unipotent subgroups contained in a finite index subgroup , together with a description of these maximal
unipotent subgroups.Comment: Comments are welcom
Small Strong Blocking Sets by Concatenation
Strong blocking sets and their counterparts, minimal codes, attracted lots of
attention in the last years. Combining the concatenating construction of codes
with a geometric insight into the minimality condition, we explicitly provide
infinite families of small strong blocking sets, whose size is linear in the
dimension of the ambient projective spaces. As a byproduct, small saturating
sets are obtained.Comment: 16 page
Construction-D lattice from Garcia-Stichtenoth tower code
We show an explicit construction of an efficiently decodable family of -dimensional lattices whose minimum distances achieve for . It improves upon the state-of-the-art construction due to Mook-Peikert (IEEE Trans.\ Inf.\ Theory, no. 68(2), 2022) that provides lattices with minimum distances . These lattices are construction-D lattices built from a sequence of BCH codes. We show that replacing BCH codes with subfield subcodes of Garcia-Stichtenoth tower codes leads to a better minimum distance. To argue on decodability of the construction, we adapt soft-decision decoding techniques of Koetter-Vardy (IEEE Trans.\ Inf.\ Theory, no.\ 49(11), 2003) to algebraic-geometric codes
Making Presentation Math Computable
This Open-Access-book addresses the issue of translating mathematical expressions from LaTeX to the syntax of Computer Algebra Systems (CAS). Over the past decades, especially in the domain of Sciences, Technology, Engineering, and Mathematics (STEM), LaTeX has become the de-facto standard to typeset mathematical formulae in publications. Since scientists are generally required to publish their work, LaTeX has become an integral part of today's publishing workflow. On the other hand, modern research increasingly relies on CAS to simplify, manipulate, compute, and visualize mathematics. However, existing LaTeX import functions in CAS are limited to simple arithmetic expressions and are, therefore, insufficient for most use cases. Consequently, the workflow of experimenting and publishing in the Sciences often includes time-consuming and error-prone manual conversions between presentational LaTeX and computational CAS formats. To address the lack of a reliable and comprehensive translation tool between LaTeX and CAS, this thesis makes the following three contributions. First, it provides an approach to semantically enhance LaTeX expressions with sufficient semantic information for translations into CAS syntaxes. Second, it demonstrates the first context-aware LaTeX to CAS translation framework LaCASt. Third, the thesis provides a novel approach to evaluate the performance for LaTeX to CAS translations on large-scaled datasets with an automatic verification of equations in digital mathematical libraries. This is an open access book
Weierstrass Semigroup, Pure Gaps and Codes on Kummer Extensions
We determine the Weierstrass semigroup at one and two totally ramified places
in a Kummer extension defined by the affine equation over , the algebraic closure of ,
where are pairwise distinct elements, and
. For an arbitrary function field, from the
knowledge of the minimal generating set of the Weierstrass semigroup at two
rational places, the set of pure gaps is characterized. We apply these results
to construct algebraic geometry codes over certain function fields with many
rational places.Comment: 24 page
Explicit Riemann-Roch spaces in the Hilbert class field
Let be a finite field, and two curves over ,
and an unramified abelian cover with Galois group . Let
be a divisor on and its pullback on . Under mild conditions the
linear space associated with is a free -module. We study
the algorithmic aspects and applications of these modules
Goppa codes over Edwards curves
Given an Edwards curve, we determine a basis for the Riemann-Roch space of
any divisor whose support does not contain any of the two singular points. This
basis allows us to compute a generating matrix for an algebraic-geometric Goppa
code over the Edwards curve.Comment: 7 pages, 1 figur
Uniform existential definitions of valuations in function fields in one variable
We study function fields of curves over a base field which is either a
global field or a large field having a separable field extension of degree
divisible by . We show that, for any such function field, Hilbert's 10th
Problem has a negative answer, the valuation rings containing are uniformly
existentially definable, and finitely generated integrally closed
-subalgebras are definable by a universal-existential formula. In order to
obtain these results, we develop further the usage of local-global principles
for quadratic forms in function fields to definability of certain subrings. We
include a first systematic presentation of this general method, without
restriction on the characteristic.Comment: 57 pages, preprin
Lower Rate Bounds for Hermitian-Lifted Codes for Odd Prime Characteristic
Locally recoverable codes are error correcting codes with the additional
property that every symbol of any codeword can be recovered from a small set of
other symbols. This property is particularly desirable in cloud storage
applications. A locally recoverable code is said to have availability if
each position has disjoint recovery sets. Hermitian-lifted codes are
locally recoverable codes with high availability first described by Lopez,
Malmskog, Matthews, Pi\~nero-Gonzales, and Wootters. The codes are based on the
well-known Hermitian curve and incorporate the novel technique of lifting to
increase the rate of the code. Lopez et al. lower bounded the rate of the codes
defined over fields with characteristic 2. This paper generalizes their work to
show that the rate of Hermitian-lifted codes is bounded below by a positive
constant depending on when for any odd prime
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