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Distributed Matrix Multiplication With Straggler Tolerance Using algebraic function fields

IEEE TRANSACTIONS ON INFORMATION THEORY 2025年第2期71卷 996-1006页

作者： Fidalgo-Diaz, Adrian Martinez-Penas, Umberto Univ Valladolid Math Res Inst IMUVa Valladolid 47002 Spain

The problem of straggler mitigation in distributed matrix multiplication (DMM) is considered for a large number of worker nodes and a fixed small finite field. Polynomial codes and matdot codes are generalized by making use of algebraic function fields (i.e., algebraic functions over an algebraic curve) over a finite field. The construction of optimal solutions is translated to a combinatorial problem on the Weierstrass semigroups of the corresponding algebraic curves. Optimal or almost optimal solutions are provided. These have the same computational complexity per worker as classical polynomial and matdot codes, and their recovery thresholds are almost optimal in the asymptotic regime (growing number of workers and a fixed finite field).

关键词： Codes Polynomials Galois fields Prevention and mitigation Encoding Costs Resistance Interpolation Computer architecture Vectors algebraic geometry codes algebraic function fields distributed matrix multiplication numerical semigroups straggler tolerance

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algebraic function fields over finite fields with many rational places

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IEEE TRANSACTIONS ON INFORMATION THEORY 1995年第6期41卷 1548-1563页

作者： Garcia, A Stichtenoth, H UNIV ESSEN GESAMTHSCH FACHBEREICH MATH & INFORMAT 6 D-45117 ESSEN GERMANY

algebraic function fields (or equivalently, algebraic curves) provide a useful tool for coding theory (for instance, algebraic-geometric codes and trace codes), but also for other branches of information theory, In these applications, the number of rational places of a function field plays a crucial role, One is particularly interested in function fields having a large number of rational places, After a short introduction into the mathematical theory of algebraic functions, the paper gives a survey of old and new results on the number of rational places of function fields.

关键词： algebraic function fields algebraic curves geometric Goppa codes AG codes trace codes m-sequences bounds for the number of rational places bounds for the code parameters

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Excellent nonlinear codes from algebraic function fields

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IEEE TRANSACTIONS ON INFORMATION THEORY 2005年第11期51卷 4044-4046页

作者： Stichtenoth, H Xing, CP Univ Disburg Essen Dept Math D-45117 Essen Germany Sabanci Univ MDBF TR-34956 Istanbul Turkey Natl Univ Singapore Dept Math Singapore 117543 Singapore

The Gilbert-Varshamov (GV) bound for asymptotic families of codes over F-q has been improved by Tsfasman, Vladut, and Zink (TVZ) in 1982, and only recently further improvements have been obtained by Xing, Elkies, and Niederreiter-Ozbudak, by considering also nonlinear codes. These improvements involve higher derivations in function fields and are very computational. We give in this correspondence a much simpler proof for those improvements. Our construction of asymptotically good nonlinear codes is very similar to Goppa's construction of algebraic-geometry codes.

关键词： algebraic function fields algebraic-geometry codes asymptotic bounds Tsfasman-Vladut-Zink (TVZ) bound

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Low increasing tower of algebraic function fields and bilinear complexity of multiplication in any extension of F_q

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FINITE fields AND THEIR APPLICATIONS 2003年第4期9卷 472-478页

作者： Ballet, S UNiv Polynesie Francaise Dept Math BP 6570 F-98702 Faaa Tahiti France

From the existence of a tower of algebraic function fields with more steps than the Garcia-Stichtenoth tower, we improve upper bounds on the bilinear complexity of multiplication in all extensions of the finite field ... 详细信息

关键词： bilinear complexity finite fields algebraic function fields algebraic curves

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A probabilistic approach to analytic arithmetic on algebraic function fields

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MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY 2005年第1期139卷 1-26页

作者： Arrattia, R Barbour, AR Tavaré, S Univ So Calif Dept Math Los Angeles CA 90089 USA Univ Zurich Abt Angew Math CH-8057 Zurich Switzerland

Knopfmacher [13] introduced the idea of an additive arithmetic semigroup as a general setting for an algebraic analogue of number theory. Within his framework, Zhang [19] showed that the asymptotic distribution of the values taken by additive functions closely resembles that found in classical number theory, in as much as there are direct analogues of the Erdos-Wintner and Kubilius Main Theorems. In this paper. we use probabilistic arguments to show that similar theorems, and their functional counterparts, can be proved in a much wider class of decomposable combinatorial structures.

关键词： .Soc classical probabilistic setting analytic taken asymptotic analogue proved Theorems decomposable algebraic function fields Knopfmacher arithmetic semigroup number theory additive functions Erdos–Wintner Kubilius combinatorial structures

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SUMS OF SQUARES IN algebraic function fields OVER A COMPLETE DISCRETELY VALUED FIELD

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PACIFIC JOURNAL OF MATHEMATICS 2014年第2期267卷 257-276页

作者： Becher, Karim Johannes Grimm, David Van Geel, Jan Univ Antwerp Dept Wiskunde & Informat B-2020 Antwerp Belgium Univ Konstanz Zukunftskolleg D-78457 Constance Germany Ecole Polytech Fed Lausanne EPFL SB MATHGEOM CSAG CH-1015 Lausanne Switzerland Univ Ghent Vakgrp Wiskunde B-9000 Ghent Belgium

A recently found local-global principle for quadratic forms over function fields of curves over a complete discretely valued field is applied to the study of quadratic forms, sums of squares, and related field invaria... 详细信息

关键词： isotropy local-global principle real field sums of squares u-invariant Pythagoras number valuation algebraic function fields

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Witt equivalence of algebraic function fields over real closed fields

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MATHEMATISCHE ZEITSCHRIFT 2002年第2期242卷 323-345页

作者： Koprowski, P Uniwersytet Slaski Inst Matemat PL-40007 Katowice Poland

We examine the conditions for two algebraic function fields over real closed fields to be Witt equivalent. We show that there are only two Witt classes of algebraic function fields with a fixed real closed field of constants: real and non-real ones. The first of them splits further into subclasses corresponding to the tame equivalence. This condition has a natural interpretation in terms of both: orderings (the associated Harrison isomorphism maps 1-pt fans onto 1-pt fans), and geometry and topology of associated real curves (the bijection of points is a homeomorphism and these two curves have the same number of semi-algebraically connected components). Finally, we derive some immediate consequences of those theorems. In particular we describe all the Witt classes of algebraic function fields of genus 0 and 1 over the fixed real closed field.

关键词： Witt equivalence algebraic function fields algebraic curves

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A Sparse Modular GCD Algorithm for Polynomials over algebraic function fields 07

A Sparse Modular GCD Algorithm for Polynomials over Algebrai...

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20th International Symposium on Symbolic and algebraic Computation

作者： Javadi, Seyed Mohammad Mahdi Monagan, Michael Simon Fraser Univ Sch Comp Sci Burnaby BC V5A 1S6 Canada

ISBN: (纸本)9781595937438

We present a first sparse modular algorithm for computing a greatest common divisor of two polynomials f(1), f(2) is an element of L[x] where L is an algebraic function field in k >= 0 parameters with r >= 0 field extensions. Our algorithm extends the dense algorithm of Monagan and van Hoeij from 2004 to support multiple field extensions and to be efficient when the gcd is sparse. Our algorithm is an output sensitive Las Vegas algorithm. We have implemented our algorithm in Maple. We provide timings demonstrating the efficiency of our algorithm compared to that of Monagan and van Hoeij and with a primitive fraction-free Euclidean algorithm for both dense and sparse gcd problems.

关键词： gcd algorithms sparse interpolation algebraic function fields

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HENSLEY'S PROBLEM FOR function fields

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INTERNATIONAL JOURNAL OF NUMBER THEORY 2012年第2期8卷 507-524页

作者： Wang, Julie Tzu-Yueh Acad Sinica Inst Math Taipei 10617 Taiwan

Buchi's square problem asks if there exists a positive integer M such that all x(1), ..., x(M) is an element of Z satisfying the equations x(r-2)(2) - 2x(r-1)(2) + x(r)(2) = 2 for all 3 <= r <= M must also satisfy x(r)(2) = (x + r)(2) for some integer x and for all 1 <= r <= M. Hensley's problem asks if there exists a positive integer M such that, for any integers nu and a, if ( nu + r)(2) - a is a square for all 1 <= r <= M, then a = 0. It is not difficult to see that a positive answer to Hensley's problem implies a positive answer to Buchi's square problem. One can ask a more general version of Hensley's problem by replacing the square power by an nth power for any integer n >= 2 which is called Hensley's problem for nth powers. In this paper, we will study Hensley's problem for nth powers over function fields of any characteristic.

关键词： Hilbert's Tenth Problem Buchi problem Hensley's problem algebraic function fields ABC theorem

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ON THE k-ERROR LINEAR COMPLEXITY OF SEQUENCES FROM function fields

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BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY 2020年第2期102卷 342-352页

作者： Zhou, Yuhui Han, Yuhui Ding, Yang Shanghai Univ Dept Math Shanghai Peoples R China

The linear complexity and the error linear complexity are two important security measures for stream ciphers. We construct periodic sequences from function fields and show that the error linear complexity of these periodic sequences is large. We also give a lower bound for the error linear complexity of a class of nonperiodic sequences.

关键词： sequence linear complexity error linear complexity algebraic function fields local expansion

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