6 research outputs found
Torsion Limits and Riemann-Roch Systems for Function Fields and Applications
The Ihara limit (or -constant) 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 -torsion points. We capture this by
the 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 non-trivial lower- and 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 areas such as coding theory, arithmetic secret sharing and multiplication
complexity of finite fields etc. 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.Comment: Accepted for publication in IEEE Transactions on Information Theory.
This is an extended version of our paper in Proceedings of 31st Annual IACR
CRYPTO, Santa Barbara, Ca., USA, 2011. The results in Sections 5 and 6 did
not appear in that paper. A first version of this paper has been widely
circulated since November 200
Improvements on parameters of one-point AG codes from Hermitian curves
10.1109/18.979330IEEE Transactions on Information Theory482535-537IETT
New examples of maximal curves with low genus
We investigate the Jacobian decomposition of some algebraic curves over
finite fields with genus , and . As a corollary, explicit equations
for curves that are either maximal or minimal over the finite field with
elements are obtained for infinitely many 's. Lists of small 's for which
maximality holds are provided. In some cases we describe the automorphism group
of the curve