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Let X be a smooth, projective and geometrically connected curve defined over a finite field ${\mathbb {F}}_q$ of characteristic p different from $2$ and $S\subseteq X$ a subset of closed points. Let $\overline {X}$ and $\overline {S}$ be their base changes to an algebraic closure of ${\mathbb {F}}_q$ . We study the number of $\ell $ -adic local sys...
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Imagine three Rubik's Cubes arranged in a row, linked such that any move on the left or right cube also applies to the middle one, and moves on the middle cube affect all three simultaneously. If each cube is individually solvable, can all three be solved together in this configuration? This question motivates a natural generalization of the Lights...
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Dedicated to the memory of Vladimir Voevodsky with admiration and gratitude. Abstract. The famous Bloch-Kato conjecture implies that for a field F containing a primitive pth root of unity, the cohomology ring of the absolute Galois group G F of F with Fp coefficients is generated by degree one elements. We investigate other groups with this propert...
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Charles et al (J Cryptol 22(1):93–113, 2009) explained how one can construct hash functions using expander graphs in which it is hard to find paths between specified vertices. The set of solutions to the classical Markoff equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \use...
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We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over $\mathbb{F}_q$ . Among other results, this allows us to prove that the $\mathbb{Q}$ -vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of t...
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The problem of determining the conditions under which a random rectangular matrix is of full rank is a fundamental question in random matrix theory, with significant implications for coding theory, cryptography, and combinatorics. In this paper, we study the probability of full rank for a K×N random matrix over the finite field Fq, where q is a pri...
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We study numerical integration over bounded regions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^s$$\end{document}, \documentclass[12pt]{minimal} \use...
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Blockchain technology offers a robust framework for integration with the Internet of Things (IoT), enhancing interoperability, security, privacy, and scalability in modern technological ecosystems. However, traditional cryptographic protocols used in blockchain systems are increasingly vulnerable to quantum attacks due to advancements in quantum co...
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Irreducible symplectic varieties are higher-dimensional analogues of K3 surfaces. In this paper, we prove the finiteness of twists of irreducible symplectic varieties for a fixed finite field extension of characteristic 0. The main ingredient of the proof is the cone conjecture for irreducible symplectic varieties, which was proved by Markman and A...
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The Pascal matrix, which is related to Pascal's triangle, appears in many places in the theory of uniform distribution and in many other areas of mathematics. Examples are the construction of low-discrepancy sequences as well as normal numbers or the binomial transforms of Hankel matrices. Hankel matrices which are defined by Catalan numbers and re...
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Let \(\mathbb{F}_{q}\) be a finite field, where \(q\) is an odd prime such that \(q>3\). Let \(f\left(t\right) =t^{3}-t\) \(\in \mathbb{F}_{q}\left[ t\right]\) be a polynomial of degree 3. For \(\lambda \neq 0\) in \(\mathbb{F}_{q}\), consider families of elliptic curves \(\left\{ E_{\lambda }\right\} _{\lambda \in \mathbb{F}_{q}^{\ast}}\) and \(\l...
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It is well-known that functions over finite fields play a crucial role in designing substitution boxes (S-boxes) in modern block ciphers. In order to analyze the security of an S-box, recently, three new tables have been introduced: the Extended Boomerang Connectivity Table (EBCT), the Lower Boomerang Connectivity Table (LBCT), and the Upper Boomer...
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Call a curve $C \subset \mathbb{P}^2$ defined over $\mathbb{F}_q$ transverse-free if every line over $\mathbb{F}_q$ intersects $C$ at some closed point with multiplicity at least 2. In 2004, Poonen used a notion of density to treat Bertini Theorems over finite fields. In this paper we develop methods for density computation and apply them to estima...
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It is well-known (cf. Weil, Gérardin’s works) that there are two different Weil representations of a symplectic group over an odd finite field. By a twisted action, we show that they can be reorganized into a representation of a projective symplectic similitude group. We also discuss the even field case by following Genestier–Lysenko and Gurevich–H...
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Let $X$ be a smooth projective variety of dimension $d$ over an arbitrary base field $k$ and $CH^n(X)_{\mathbb Q}$ be the $\mathbb Q$-vector space of codimension $n$ algebraic cycles of $X$ modulo rational equivalence, $1\leq n \leq d$. Consider the $\mathbb Q$-vector subspaces $CH^n(X)_{\mathbb Q} \supseteq CH^n_{\mathrm{alg}}(X)_{\mathbb Q} \sups...
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Let Fq be a finite field, and let n be a positive integer such that gcd(q,n)=1. The irreducible factors of xn−1 and xn−λ are fundamental concepts with wide applications in cyclic codes and constacyclic codes. Furthermore, the number of irreducible factors of xn−1 and xn−λ is useful in many computational problems involving cyclic codes and constacyc...
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Due to the operational efficiency and lower computational costs of the Chebyshev polynomial compared to ECC, this chaotic system has attracted widespread attention in public key cryptography. However, the single recurrence coefficient limitation and inherent short-period flaw, often render the Chebyshev polynomials cryptosystem ineffective against...
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Let \(\mathbb {F}_q\) be the finite field with q elements, where q is a power of a prime p. Given a monic polynomial \(f \in \mathbb {F}_q[x]\) that is not divisible by x, there exists a positive integer \(e=e(f)\) such that f(x) divides the binomial \(x^e-1\) and e is minimal with this property. The integer e is commonly known as the order of f an...
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Let $$\mathbb {F}_q$$ F q be the finite field with q elements and $$\mathbb {F}_q[x_1,\ldots , x_n]$$ F q [ x 1 , … , x n ] the ring of polynomials in n variables over $$\mathbb {F}_q$$ F q . In this paper we consider permutation polynomials and local permutation polynomials over $$\mathbb {F}_q[x_1,\ldots , x_n]$$ F q [ x 1 , … , x n ] , which def...
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Secure Multi-Party Computation (MPC) is an important enabling technology for data privacy in modern distributed applications. We develop a new type theory to automatically enforce correctness,confidentiality, and integrity properties of protocols written in the \emph{Prelude/Overture} language framework. Judgements in the type theory are predicated...
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Continuous variable quantum key distribution bears the promise of simple quantum key distribution directly compatible with commercial off the shelf equipment. However, for a long time its performance was hindered by the absence of good classical postprocessing capable of distilling secret-keys in the noisy regime. Advanced coding solutions in the p...
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We describe a deterministic process to associate a practical, permanent label to isomorphism classes of abelian varieties defined over finite fields and the polarizations they admit, for use in the mathematical literature.
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Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document} be the finite field with q elements, \documentclass[12pt]{minimal} \usepackage{amsmath...
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In this paper, we obtain a finite field analog for a transformation satisfied by 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1$$\end{document}-classical hypergeo...
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Given the finite field $\mathbb{F}_{q}$, for a prime power $q$, in this paper we present a way of constructing spreads of $\mathbb{F}_{q}^{n}$. They will arise as orbits under the action of an Abelian non-cyclic group. First, we construct a family of orbit codes of maximum distance using this group, and then we complete each of these codes to achie...
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If X is a variety over a number field, Annette Huber has defined a category of “horizontal” (or “almost everywhere unramified”) \ell -adic complexes and \ell -adic perverse sheaves on X . For such objects, the notion of weights makes sense (in the sense of Deligne), just as in the case of varieties over finite fields. However, contrary to what happ...
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In this paper, we provide two new methods of constructing entanglement-assisted quantum error-correcting (EAQEC) codes by using the LCD codes decomposition of linear codes. We first construct a class of maximal entanglement EAQEC maximum distance separable codes via the LCD codes decomposition of generalized Reed–Solomon (GRS) codes over finite fie...
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CairoZero is a programming language for running decentralized applications (dApps) at scale. Programs written in the CairoZero language are compiled to machine code for the Cairo CPU architecture and cryptographic protocols are used to verify the results of execution efficiently on blockchain. We explain how we have extended the CairoZero compiler...
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We present a new application of multi-orbit cyclic subspace codes to construct large optical orthogonal codes, with the aid of the multiplicative structure of finite fields extensions. This approach is different from earlier approaches using combinatorial and additive (character sum) structures of finite fields. Consequently, we immediately obtain...
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Recently, the first author as well as the second author with Ono, Pujahari, and Saikia determined the limiting distribution of values of certain finite field ${_2F_1}$ and ${_3F_2}$ hypergeometric functions. These hypergeometric values are related to Frobenius traces of elliptic curves and their limiting distribution is determined using connections...
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We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage...
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Using basic properties of perverse sheaves, we give new upper bounds for compactly supported Betti numbers for arbitrary affine varieties in $\mathbb{A}^n$ defined by $r$ polynomial equations of degrees at most $d$. As arithmetic applications, new total degree bounds are obtained for zeta functions of varieties and L-functions of exponential sums o...
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We present a novel proof that the maximum number of sets with 4 properties for 12 cards is 14 using the geometry of the finite field F_3^4, number theory, combinatorics, and graph theory. We also present several computer algorithms for finding the maximum number of sets. In particular, we show a complete set solver that iterates over all possible b...
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We study the Random Access Problem in DNA storage, which addresses the challenge of retrieving a specific information strand from a DNA-based storage system. Given that $k$ information strands, representing the data, are encoded into $n$ strands using a code. The goal under this paradigm is to identify and analyze codes that minimize the expected n...
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In this paper, we mainly consider arithmetic properties of the cyclotomic matrix $B_p(k)=\left[J_p(\chi^{ki},\chi^{kj})^{-1}\right]_{1\le i,j\le (p-1-k)/k}$, where $p$ is an odd prime, $1\le k<p-1$ is a divisor of $p-1$, $\chi$ is a generator of the group of all multiplicative characters of the finite field $\mathbb{F}_p$ and $J_p(\chi^{ki},\chi^{k...
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Let $\eta_{g}(n) $ be the smallest cardinality that $A\subseteq {\mathbb Z}$ can have if $A$ is a $g$-difference basis for $[n]$ (i.e, if, for each $x\in [n]$, there are {\em at least} $g$ solutions to $a_{1}-a_{2}=x$ ). We prove that the finite, non-zero limit $\lim\limits_{n\rightarrow \infty}\frac{\eta_{g}(n)}{\sqrt{n}}$ exists, answering a ques...
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In this paper, we obtain a class of bijections between the projective geometry $PG(n-1,q)$ and the set of roots of unity $\mu_{\frac{q^n-1}{q-1}}$ in finite field $\mathbb{F}_{q^n}$ for an arbitrary integer $n\geq 2$ and any basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$. This generalizes the well-studied M\"obius transformations for $n=2$ and a...
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We construct $\mathbb{Z}_p$-lattices and $\mathbb{F}_q[\![t]\!]$-lattices from cyclic $(f,\sigma)$-codes over finite chain rings, employing quotients of natural nonassociative orders and principal left ideals in carefully chosen nonassociative algebras. This approach generalizes the classical Construction A that obtains $\mathbb{Z}$-lattices from l...
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We present an algorithm that, for every fixed genus g, will enumerate all hyperelliptic curves of genus g over a finite field k of odd characteristic in quasilinear time; that is, the time required for the algorithm is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsb...
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Restricted by the environment and hardware equipment resources, existing chaotic systems have shortcomings such as low complexity, low randomness, and chaotic degradation phenomena, which in turn cause the security risks of chaotic image encryption algorithms. To overcome these issues, this paper proposes a method for the construction of a LE-contr...
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It was shown by Boukerrou et al. [IACR Trans. Symmetric Cryptol. 1 (2020), 331–362] that the F-boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic) of perfect nonlinear functions is 0 on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepacka...
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Contraction$^*$-depth is considered to be one of the analogues of graph tree-depth in the matroid setting. In this paper, we investigate structural properties of contraction$^*$-depth of matroids representable over finite fields and rationals. In particular, we prove that the obstructions for contraction$^*$-depth for these classes of matroids are...
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For $X$ a finite category and $F$ a finite field, we study the additive image of the functor $\operatorname{H}_0(-,F) \colon \operatorname{rep}(X, \mathbf{Top}) \to \operatorname{rep}(X, \mathbf{Vect}_F)$, or equivalently, of the free functor $\operatorname{rep}(X, \mathbf{Set}) \to \operatorname{rep}(X, \mathbf{Vect}_F)$. We characterize all finit...
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In (Finite Fields Their Appl. 46, 38–56 2017), Wu et al. defined the notion of quasi-multiplicative (QM) equivalence among permutation polynomials. Other than showing thoroughly, there is no efficient approach to determine whether two given permutation polynomials are QM equivalent or not. This paper provides new results to determine QM equivalence...
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Let q be a power of a prime p , $$\mathbb {F}_q$$ F q be the finite field with q elements, and $$\mathbb {F}_q[x_1,\ldots , x_n]$$ F q [ x 1 , … , x n ] be the ring of polynomials in n variables over $$\mathbb {F}_q$$ F q . The construction and study of local permutation polynomials of $$\mathbb {F}_q[x_1,\ldots , x_n]$$ F q [ x 1 , … , x n ] is re...
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Everlasting (EL) privacy offers an attractive solution to the Store-Now-Decrypt-Later (SNDL) problem, where future increases in the attacker's capability could break systems which are believed to be secure today. Instead of requiring full information-theoretic security, everlasting privacy allows computationally-secure transmissions of ephemeral se...
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We prove a converse theorem for the case of quasi-split non-split even special orthogonal groups over finite fields. There are two main difficulties which arise from the outer automorphism and non-split part of the torus. The outer automorphism is handled similarly to the split case, while new ideas are developed to overcome the non-split part of t...
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We investigate the descriptive set-theoretic complexity of the solvability of a Borel family of linear equations over a finite field. Answering a question of Thornton, we show that this problem is already hard, namely $\Sigma^1_2$-complete. This implies that the split between easy and hard problems is at a different place in the Borel setting than...
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Let F = F|K be a function field over an algebraically closed constant field K of positive characteristic p. For a K-automorphism group G of F, the invariant of G is the fixed field FG of G. If F has transendency degree 1 (i.e. F is the function field of an irreducible curve) and FG is rational, then each generator of FG uniquely determines FG and i...
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With the rise of meta-universe, the application of 3D models in various fields has garnered significant attention, for they may furnish more vivid and detailed representations. However, many problems lie in securing the transmission of 3D model data, as the tampered possibility increases while in transit. Therefore, protecting 3D model data is cruc...
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Let $\xi\in\mathbb{F}_{q^m}$ be an $r$-primitive $k$-normal element over $\mathbb{F}_q$, where $q$ is a prime power and $m$ is a positive integer. The minimal polynomial of $\xi$ is referred to be the $r$-primitive $k$-normal polynomial of $\xi$ over $\mathbb{F}_q$. In this article, we study the existence of an $r$-primitive $k$-normal polynomial o...
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Network coding enhances performance in network communications and distributed storage by increasing throughput and robustness while reducing latency. Batched Sparse (BATS) codes are a class of capacity-achieving network codes, but their practical applications are hindered by their structure, computational intensity, and power demands of finite fiel...
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We compute $\mathrm{Ext}^{1}_B(\chi_1,\chi_2)$ between two characters $\chi_1,\chi_2$ of a Borel subgroup $B$ of a split reductive group $G$ over a finite field $\mathbb{F}_q,$ and make an application to the calculation of $\mathrm{Ext}^1_G(\pi_1,\pi_2)$ between principal series representations $\pi_1,\pi_2$ of $G(\mathbb{F}_q).
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Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$, where $p$ is a prime number and $m$ is a positive integer. Self-dual constacyclic codes of length \( p^s \) over \( \frac{\mathbb{F}_{p^m}[u]}{\langle u^3 \rangle} \) exist only when \( p = 2 \). In this work, we classify and enumerate all self-dual cyclic codes of length \( 2^s \) over...
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Let $P(t),Q(t)\in \mathbb{Q}(t)$ be rational functions such that $P(t),Q(t)$ and the constant function $1$ are linearly independent over $\mathbb{Q}$, we prove an asymptotic formula for the number of the corner configurations $(x_1,x_2),(x_1+P(y),x_2),(x_1,x_2+Q(y))$ in the subsets of $\mathbb{F}_p^2$.
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We focus on two aspects of cyclic orbit codes: invariants under equivalence and quasi-optimality. Regarding the first aspect, we establish a connection between the codewords of a cyclic orbit code and a certain linear set on the projective line. This allows us to derive new bounds on the parameters of the code. In the second part, we study a partic...
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In 1980 Ian G. Macdonald established an explicit bijection between the isomorphism classes of the irreducible representations of ${\mathrm{GL}}_n(k)$, where $k$ is a finite field, and inertia equivalence classes of admissible tamely ramified $n$-dimensional Weil-Deligne representations of $W_F$, where $F$ is a non-archimedean local field with resid...
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In this paper, we consider a version of the bias conjecture for second moments in the setting of elliptic curves over finite fields whose trace of Frobenius lies in a fixed arithmetic progression. Contrary to the classical setting of reductions of one-parameter families over the rationals, where it is conjectured by Steven J. Miller that the bias i...
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Let $E$ be a subset of the the affine plane over a finite field $\mathbb{F}_q$. We bound the size of the subgroup of $SL_2(\mathbb{F}_q)$ that preserves $E$. As a consequence, we show that if $E$ has size $\ll q^\alpha$ and is preserved by $\gg q^\beta$ elements of $SL_2(\mathbb{F}_q)$ with $\beta\geq 3\alpha/2$, then $E$ is contained in a line. Th...
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We study the distinction of the Steinberg representation of a split reductive group $G$ with respect to a split symmetric subgroup $H \subset G$. We do that both over a $p$ adic field and over a finite field. We relate these distinction problems to problems about determining the existence of a non zero harmonic function on certain hyper graphs rela...
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In this article, for a finite field $\mathbb{F}_q$ and a natural number $l,$ let $\mathcal{R}:=\mathbb{F}_q^l.$ Firstly, for an automorphism $\Theta$ of $\mathcal{R},$ a $\Theta$-derivation $\Delta_\Theta$ of $\mathcal{R}$ and $\mathbf{a}\in \mathcal{R}^\times,$ we study $(\Theta, \Delta_\Theta, \mathbf{a})$-cyclic codes over the skew polynomial ri...
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This paper characterizes Goppa codes of certain maximal curves over finite fields defined by equations of the form $y^n = x^m + x$. We investigate Algebraic Geometric and quantum stabilizer codes associated with these maximal curves and propose modifications to improve their parameters. The theoretical analysis is complemented by extensive simulati...
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High throughput broadcasting systems requires efficient broadcasting that can broadcast the information source symbols with optimal rate (i.e., maximizing the usage of broadcast channel which exploits the broadcast nature of the wireless medium through index coding). Index coding is the elegant and beautiful idea, which is used to transform the inf...
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Let q = p<sup>r</sup> be a prime power, F<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> q </sub> be the finite field of order q and f ( x ) be a monic polynomial in F<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> q </sub>[ x ]. Set A := F<sub xmlns:mml...
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In the distribution network under the background of disaster, the distribution terminal embodies the characteristics of disorder. Therefore, when the distribution terminal node accesses the backup network, there will be congestion, delay and other phenomena. In order to solve this problem, this paper proposes a distribution terminal access mechanis...
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For a Henselian discrete valued field $K$ of characteristic $p>0$, Kato defined a ramification filtration $\{{\rm fil}_nH^q(K,\mathbb Q_p/\mathbb Z_p(q-1))\}_{n \ge 0}$ on $H^q(K,\mathbb Q_p/\mathbb Z_p(q-1))$. One can also define a ramification filtration on $H^q(U,\mathbb Z/p^m(q-1))$ using the local Kato-filtration, where $U$ is the complement o...
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Arithmetic-geometric mean sequences were already studied over real and complex numbers, and recently, Michael J. Griffin, Ken Ono, Neelam Saikia and Wei-Lun Tsai considered them over finite fields $\mathbb{F}_q$ such that $q \equiv 3 \pmod 4$. In this paper, we extend the definition of arithmetic-geometric mean sequences over $\mathbb{F}_q$ such th...
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In this study, the positive solutions of the Diophantine equation D : x 2 − σ 2 + 4 y 2 − (2σ − 2) x − 2σ 4 + 8σ 2 y − σ 6 − 4σ 4 + σ 2 − 2σ − 3 = 0 on the set Z are investigated, along with some recurrence relations that provide the relationships among these solutions. In addition, solutions of the Diophantine equation D in terms of generalized Fi...
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We continue the formalization of field theory in Mizar. Here we prove existence and uniqueness of finite fields by constructing the splitting field of the polynomial X (p ⁿ ) − X over the prime field of a field with characteristic p . We also define the Frobenius morphism and show that the automorphisms of a field with p ⁿ elements are exactly the...
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Our work addresses the well-known open problem of distributed computing of bilinear functions of two correlated sources ${\bf A}$ and ${\bf B}$. In a setting with two nodes, with the first node having access to ${\bf A}$ and the second to ${\bf B}$, we establish bounds on the optimal sum-rate that allows a receiver to compute an important class of...
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This paper gives an insight to the Galois theory and discusses its applications in both pure and applied mathematics. First, the Fundamental theorem of Galois theory is applied to compute the Galois groups of polynomials and to prove the non-existence of a formula for solving a polynomial equation in rational coefficients having degree n ≥ 5. Then...
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Arithmetic-geometric mean sequences were already studied over real and complex numbers, and recently, Michael J. Griffin, Ken Ono, Neelam Saikia and Wei-Lun Tsai considered them over finite fields Fq such that q ≡ 3 (mod 4). In this paper, we extend the definition of arithmetic-geometric mean sequences over Fq such that q ≡ 5 (mod 8). We explain th...
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A bstract We present the two-loop helicity amplitudes contributing to the next-to-next-to-leading order QCD predictions for W -boson production in association with two photons at the Large Hadron Collider. We derived compact analytic expressions for the two-loop amplitudes in the leading colour limit, and provide numerical results for the subleadin...
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Strong algebraic proof systems such as IPS (Ideal Proof System; Grochow-Pitassi [GP18]) offer a general model for deriving polynomials in an ideal and refuting unsatisfiable propositional formulas, subsuming most standard propositional proof systems. A major approach for lower bounding the size of IPS refutations is the Functional Lower Bound Metho...
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In this work, we explore the use of maximal elements in generalized Weierstrass semigroups and their relationship with pure gaps, extending the results in Castellanos et al. [J Pure Appl Algebra 228(4):107513, 2024]. We provide a method to completely determine the set of pure gaps at several rational places in a function field F over a finite field...
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We study the behavior of the multiplicative inverse function (which plays an important role in cryptography and in the study of finite fields), with respect to a recently introduced generalization of almost perfect nonlinearity (APNness), called kth-order sum-freedom, that extends a classic characterization of APN functions, and has also some relat...
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Studying the generalized Hamming weights of linear codes is a significant research area within coding theory, as it provides valuable structural information about the codes and plays a crucial role in determining their performance in various applications. However, determining the generalized Hamming weights of linear codes, particularly their weigh...
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In this article, for the finite field $\mathbb{F}_q$, we show that the $\mathbb{F}_q$-algebra $\mathbb{F}_q[x]/\langle f(x) \rangle$ is isomorphic to the product ring $\mathbb{F}_q^{\deg f(x)}$ if and only if $f(x)$ splits over $\mathbb{F}_q$ into distinct factors. We generalize this result to the quotient of the polynomial algebra $\mathbb{F}_q[x_...
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A large number of algorithms solving Coding Theory problems involve operations on vectors over finite fields. The use of extended CPU registers and instructions is suitable for the optimization of these algorithms. Current work presents the Neon instruction set for the ARM architectures used in Apple's M series of processors. A method for their app...
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The original problem that serves as a basis for this project comes from an American contest (PUMaC, 2014) regarding the maximum amount of enclosed spaces given a limited number of cuts on an infinite plane. In this study, we explore the same problem and extend it in the context of m dimensions given n (m-1) dimensional cuts using the recursive rela...
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This paper proves that the Nagata automorphism over a finite field can be mimicked by a tame automorphism which is a composition of four elementary automorphisms. By investigating the sign of the permutations induced by the above elementary automorphisms, one can see that if the Nagata automorphism is defined over a prime field of characteristic tw...
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We consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on an irreducible characteristic polynomial of degree $n$ and order $m$. Let $t=(q^{n}-1)/ m$. We use quadratic forms over finite fields to give the exact number of occurrences of zeros of the sequence within its least period when $t$ has q-adic weight 2....
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In this paper, we investigate the sign of permutations induced by the Anick automorphism and the Nagata-Anick automorphism over finite fields. We shall prove that if the Anick automorphism and the Nagata-Anick automorphism are defined over a prime field of characteristic two, they induce odd permutations, and otherwise, they induce even permutation...
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Consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on the irreducible characteristic polynomial of degree $d$ and order $m$. We give upper and lower bounds, and in some cases the exact values of the cardinality of the set of zeros of the sequences within its least period. We also prove that the cyclotomy bou...
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We obtain new bounds for (a variant of) the Furstenberg set problem for high dimensional flats over $\mathbb{R}^n$. In particular, let $F\subset \mathbb{R}^n$, $1\leq k \leq n-1$, $s\in (0,k]$, and $t\in (0,k(n-k)]$. We say that $F$ is a $(s,t;k)$-spread Furstenberg set if there exists a $t$-dimensional set of subspaces $\mathcal P \subset \mathcal...
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We introduce a formula for determining the number of codewords of weight 2 in cyclic codes and provide results related to the count of codewords with weight 3. Additionally, we establish a recursive relationship for binary cyclic codes that connects their weight distribution to the number of solutions of associated systems of polynomial equations....
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As technology increasingly permeates our daily lives, the role of cryptography has become crucial, especially in the healthcare sector. Medical scans' confidentiality is paramount; unauthorized access or leakage could lead to severe privacy violations and potential misuse of personal data. Cryptography safeguards the privacy and integrity of this d...
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