NDLI: Torsion Limits and Riemann-Roch Systems for Function Fields and Applications

Content Provider	IEEE Xplore Digital Library
Author	Cascudo, I. Cramer, R. Chaoping Xing
Copyright Year	1963
Abstract	The Ihara limit (or constant) A(q) has been a central problem of study in the asymptotic theory of global function fields (or equivalently, algebraic curves over finite fields). It addresses global function fields with many rational points and, so far, most applications of this theory do not require additional properties. Motivated by recent applications, we require global function fields with the additional property that their zero class divisor groups contain at most a small number of d -torsion points. We capture this with the notion of torsion limit, a new asymptotic quantity for global function fields. It seems that it is even harder to determine values of this new quantity than the Ihara constant. Nevertheless, some nontrivial upper bounds are derived. Apart from this new asymptotic quantity and bounds on it, we also introduce Riemann-Roch systems of equations. It turns out that this type of equation system plays an important role in the study of several other problems in each of these areas: arithmetic secret sharing, symmetric bilinear complexity of multiplication in finite fields, frameproof codes, and the theory of error correcting codes. Finally, we show how our new asymptotic quantity, our bounds on it and Riemann-Roch systems can be used to improve results in these areas.
Sponsorship	IEEE Information Theory Society
Starting Page	3871
Ending Page	3888
Page Count	18
File Size	566744
File Format	PDF
ISSN	00189448
Volume Number	60
Issue Number	7
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2014-01-01
Publisher Place	U.S.A.
Access Restriction	One Nation One Subscription (ONOS)
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Cryptography Equations Poles and towers Upper bound Context Complexity theory Linear codes frameproof codes Algebraic curves jacobian torsion limit Ihara limit secret sharing complexity of multiplication
Content Type	Text
Resource Type	Article
Subject	Library and Information Sciences Information Systems Computer Science Applications

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