Arithmetic, Geometry, Cryptography and Coding Theory

by Bassa, Alp

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Full title :Arithmetic, Geometry, Cryptography and Coding Theory

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Cover -- Title page -- Contents -- Preface -- The exact limit of some cubic towers -- 1. Introduction -- 2. Various Cubic Towers -- 3. The Exact Genus and Exact Limit of Tower \GG -- 4. Conclusion -- Acknowledgment -- References -- Error-correction capability of Reed-Muller codes -- 1. Introduction -- 2. Presentation -- 3. The results -- 4. Proof of Theorem 3.1. Decoding a large number of errors -- 5. The error correction capability function -- 6. Note -- References -- Optimal and maximal singular curves -- 1. Introduction -- 2. Hallouin-Perret's approach -- 3. Number of closed points of degree 2 -- 4. Genera spectrum of maximal curves -- References -- A infinite class of Kasami functions that are not APN infinitely often -- 1. Introduction -- 2. Preliminaries -- 3. Another proof of a theorem of Janwa-Wilson -- 4. Hyperplane sections -- 5. Polynomials of Kasami Degree -- Acknowledgements -- References -- Covariant algebra of the binary nonic and the binary decimic -- 1. Introduction -- 2. Mathematical framework -- 3. Gordan's algorithm -- 4. Improvements of Gordan's algorithm -- 5. Computational aspects -- 6. Results -- Acknowledgement -- References -- Appendix A.A minimal basis for \cov{\Sn{9}} -- Appendix B.A minimal basis for \cov{\Sn{10}} -- On some bounds for symmetric tensor rank of multiplication in finite fields -- 1. Introduction -- 2. New upper bounds for the symmetric bilinear complexity -- 3. Note on some unproven bounds -- Acknowledgment -- References -- Codes from Jacobian surfaces -- 1. Introduction -- 2. Curves on abelian surfaces over finite fields -- 3. Codes from Jacobian surfaces -- 4. Some numerical results -- Acknowledgments -- References -- A new proof of a Thomae-like formulafor non hyperelliptic genus 3 curves -- 1. Introduction -- 2. Review on Aronhold sets, fundamental systems and Riemann-Jacobi formula

3. Proofs of Weber's formula -- Acknowledgment -- References -- Remarks on the Tsfasman-Boguslavsky Conjecture and higher weights of projective Reed-Muller codes -- 1. Introduction -- 2. Preliminaries -- 3. TBC for Systems of Three Polynomial Equations -- 4. Projective Reed-Muller codes and their higher weights -- 5. A Counterexample to the TBC -- Acknowledgment -- References -- Secret sharing schemes with strong multiplication and a large number of players from toric varieties -- 1. Introduction -- 2. Preliminaries -- 3. Linear secret sharing schemes with multiplication on tori -- 4. Toric surfaces and linear secret sharing schemes with strong multiplication -- References -- Field extensions and index calculus on algebraic curves -- 1. Introduction -- 2. Index calculus and the divisor class group -- 3. Known index calculus methods -- 4. The general case -- 5. Conclusion -- References -- Back Cover

This volume contains the proceedings of the 15th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory (AGCT), held at the Centre International de Rencontres Mathématiques in Marseille, France, from May 18-22, 2015. Since the first meeting almost 30 years ago, the biennial AGCT meetings have been one of the main events bringing together researchers interested in explicit aspects of arithmetic geometry and applications to coding theory and cryptography. This volume contains original research articles reflecting recent developments in the field