Name
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. |
Year | DOI | Venue |
|---|---|---|
2014 | 10.1109/TIT.2014.2314099 | Information Theory, IEEE Transactions |
Keywords | Field | DocType |
algebra,algebraic codes,error correction codes,Ihara limit,Riemann-Roch System,algebraic curves over finite field,arithmetic secret sharing,asymptotic theory,equation system,error correcting code,function fields,global function field,symmetric bilinear multiplication complexity,torsion limit,zero class divisor groups,Algebraic curves,Ihara limit,complexity of multiplication,frameproof codes,jacobian,secret sharing,torsion limit | Discrete mathematics,Finite field,Combinatorics,System of linear equations,Algebraic curve,Upper and lower bounds,Multiplication,Riemann hypothesis,Divisor,Mathematics,Bilinear interpolation | Journal |
Volume | Issue | ISSN |
60 | 7 | 0018-9448 |
Citations | PageRank | References |
8 | 0.59 | 26 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Ignacio Cascudo Pueyo | 1 | 118 | 11.41 |
| Ronald Cramer | 2 | 2499 | 178.28 |
| Chaoping Xing | 3 | 916 | 110.47 |