1,903 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology

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    We study smooth higher symmetry groups and moduli ∞\infty-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 MM 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 ∞\infty-categorical quotients by the action of the higher symmetries, extract information about the homotopy types of these moduli ∞\infty-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 U(1)\mathrm{U}(1)-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 nn-gerbes with kk-connection, comment on the relation to higher-form symmetries, and present a new String group model. We construct smooth moduli ∞\infty-stacks of higher Maxwell and Einstein-Maxwell solutions, correcting previous such considerations in the literature, and compute the homotopy groups of several moduli ∞\infty-stacks of higher U(1)\mathrm{U}(1)- 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 ∞\infty-stacks of NSNS supergravity solutions.Comment: 102 pages; comments welcom

    Several families of ternary negacyclic codes and their duals

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    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

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    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

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    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 Uq(sl2)U_q(\mathfrak{sl}_2) and the Terwilliger algebras of Grassmann graphs

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    The Clebsch--Gordan coefficients of U(sl2)U(\mathfrak{sl}_2) are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism from the universal Hahn algebra H\mathcal H into U(sl2)⊗U(sl2)U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2). Let Ω\Omega denote a finite set and 2Ω2^\Omega denote the power set of Ω\Omega. It is generally known that C2Ω\mathbb C^{2^\Omega} supports a U(sl2)U(\mathfrak{sl}_2)-module. Fix an element x0∈2Ωx_0\in 2^\Omega. By the linear isomorphism C2Ω→C2Ω∖x0⊗C2x0\mathbb C^{2^\Omega}\to \mathbb C^{2^{\Omega\setminus x_0}}\otimes \mathbb C^{2^{x_0}} given by x↦(x∖x0)⊗(x∩x0)x\mapsto (x\setminus x_0)\otimes (x\cap x_0) for all x∈2Ωx\in 2^\Omega, this induces a U(sl2)⊗U(sl2)U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)-module structure on C2Ω\mathbb C^{2^\Omega}. Pulling back via the algebra homomorphism H→U(sl2)⊗U(sl2)\mathcal H\to U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2), the U(sl2)⊗U(sl2)U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)-module C2Ω\mathbb C^{2^\Omega} forms an H\mathcal H-module. The H\mathcal H-module C2Ω\mathbb C^{2^\Omega} enfolds the Terwilliger algebra of a Johnson graph. This result connects these two seemingly irrelevant topics: The Clebsch--Gordan coefficients of U(sl2)U(\mathfrak{sl}_2) and the Terwilliger algebras of Johnson graphs. Unfortunately some steps break down in the qq-analog case. By making detours, the imperceptible connection between the Clebsch--Gordan coefficients of Uq(sl2)U_q(\mathfrak{sl}_2) and the Terwilliger algebras of Grassmann graphs is successfully disclosed in this paper.Comment: 65 page

    Z2Z4Z8\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8-Additive Hadamard Codes

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    The Z2Z4Z8\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8-additive codes are subgroups of Z2α1×Z4α2×Z8α3\mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3}, and can be seen as linear codes over Z2\mathbb{Z}_2 when α2=α3=0\alpha_2=\alpha_3=0, Z4\mathbb{Z}_4-additive or Z8\mathbb{Z}_8-additive codes when α1=α3=0\alpha_1=\alpha_3=0 or α1=α2=0\alpha_1=\alpha_2=0, respectively, or Z2Z4\mathbb{Z}_2\mathbb{Z}_4-additive codes when α3=0\alpha_3=0. A Z2Z4Z8\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8-linear Hadamard code is a Hadamard code which is the Gray map image of a Z2Z4Z8\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8-additive code. In this paper, we generalize some known results for Z2Z4\mathbb{Z}_2\mathbb{Z}_4-linear Hadamard codes to Z2Z4Z8\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8-linear Hadamard codes with α1≠0\alpha_1 \neq 0, α2≠0\alpha_2 \neq 0, and α3≠0\alpha_3 \neq 0. First, we give a recursive construction of Z2Z4Z8\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8-additive Hadamard codes of type (α1,α2,α3;t1,t2,t3)(\alpha_1,\alpha_2, \alpha_3;t_1,t_2, t_3) with t1≥1t_1\geq 1, t2≥0t_2 \geq 0, and t3≥1t_3\geq 1. Then, we show that in general the Z4\mathbb{Z}_4-linear, Z8\mathbb{Z}_8-linear and Z2Z4\mathbb{Z}_2\mathbb{Z}_4-linear Hadamard codes are not included in the family of Z2Z4Z8\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8-linear Hadamard codes with α1≠0\alpha_1 \neq 0, α2≠0\alpha_2 \neq 0, and α3≠0\alpha_3 \neq 0. Actually, we point out that none of these nonlinear Z2Z4Z8\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8-linear Hadamard codes of length 2112^{11} is equivalent to a Z2Z4Z8\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8-linear Hadamard code of any other type, a Z2Z4\mathbb{Z}_2\mathbb{Z}_4-linear Hadamard code, or a Z2s\mathbb{Z}_{2^s}-linear Hadamard code, with s≥2s\geq 2, of the same length 2112^{11}

    Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats

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    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

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    In this paper, we present a generalization of Hayden's theorem [7, Theorem 4.2] for GG-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 GG-codes over finite Frobenius rings. Furthermore, we provide the Jacobi analogue of Astumi's MacWilliams identity for GG-codes over finite Frobenius rings. Finally, we study the relation between GG-codes and its corresponding GG-lattices.Comment: 22 page
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