401 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On Chevalley group schemes over function fields: quotients of the Bruhat-Tits building by {β„˜}\{\wp\}-arithmetic subgroups

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    Let G\mathbf{G} be a reductive Chevalley group scheme (defined over Z\mathbb{Z}). Let C\mathcal{C} be a smooth, projective, geometrically integral curve over a field F\mathbb{F}. Let β„˜\wp be a closed point on C\mathcal{C}. Let AA be the ring of functions that are regular outside {β„˜}\lbrace \wp \rbrace. The fraction field kk of AA has a discrete valuation Ξ½=Ξ½β„˜:kΓ—β†’Z\nu=\nu_{\wp}: k^{\times} \rightarrow \mathbb{Z} associated to β„˜\wp. In this work, we study the action of the group G(A) \textbf{G}(A) of AA-points of G\mathbf{G} on the Bruhat-Tits building X=X(G,k,Ξ½β„˜)\mathcal{X}=\mathcal{X}(\textbf{G},k,\nu_\wp) in order to describe the structure of the orbit space G(A)\X \textbf{G}(A)\backslash \mathcal{X}. 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 Cβˆ–{β„˜}\mathcal{C} \smallsetminus \{\wp\} and on the rank of G\mathbf{G}. 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 G(A)\mathbf{G}(A) as a free product with amalgamation. We also obtain a counting of the Ξ“\Gamma-conjugacy classes of maximal unipotent subgroups contained in a finite index subgroup Ξ“βŠ†G(A)\Gamma \subseteq \mathbf{G}(A), together with a description of these maximal unipotent subgroups.Comment: Comments are welcom

    Small Strong Blocking Sets by Concatenation

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    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

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    We show an explicit construction of an efficiently decodable family of nn-dimensional lattices whose minimum distances achieve Ω(n/(log⁑n)Ρ+o(1))\Omega(\sqrt{n} / (\log n)^{\varepsilon+o(1)}) for Ρ>0\varepsilon>0. 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 Ω(n/log⁑n)\Omega(\sqrt{n/ \log n}). 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

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    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

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    We determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation ym=∏i=1r(xβˆ’Ξ±i)Ξ»iy^{m}=\prod_{i=1}^{r} (x-\alpha_i)^{\lambda_i} over KK, the algebraic closure of Fq\mathbb{F}_q, where Ξ±1,…,Ξ±r∈K\alpha_1, \dots, \alpha_r\in K are pairwise distinct elements, and gcd⁑(m,βˆ‘i=1rΞ»i)=1\gcd(m, \sum_{i=1}^{r}\lambda_i)=1. 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

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    Let K\mathbf K be a finite field, XX and YY two curves over K\mathbf K, and Y→XY\rightarrow X an unramified abelian cover with Galois group GG. Let DD be a divisor on XX and EE its pullback on YY. Under mild conditions the linear space associated with EE is a free K[G]{\mathbf K}[G]-module. We study the algorithmic aspects and applications of these modules

    Goppa codes over Edwards curves

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    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

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    We study function fields of curves over a base field KK which is either a global field or a large field having a separable field extension of degree divisible by 44. We show that, for any such function field, Hilbert's 10th Problem has a negative answer, the valuation rings containing KK are uniformly existentially definable, and finitely generated integrally closed KK-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

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    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 tt if each position has tt 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 pp when q=plq=p^l for any odd prime pp
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