119 research outputs found
Transitive and self-dual codes attaining the Tsfasman-Vladut-Zink bound
A major problem in coding theory is the question of whether the class of cyclic codes is asymptotically good. In this correspondence-as a generalization of cyclic codes-the notion of transitive codes is introduced (see Definition 1.4 in Section I), and it is shown that the class of transitive codes is asymptotically good. Even more, transitive codes attain the Tsfasman-Vladut-Zink bound over F-q, for all squares q = l(2). It is also shown that self-orthogonal and self-dual codes attain the Tsfasman-Vladut-Zink bound, thus improving previous results about self-dual codes attaining the Gilbert-Varshamov bound. The main tool is a new asymptotically optimal tower E-0 subset of E-1 subset of E-2 subset of center dot center dot center dot of function fields over F-q (with q = l(2)), where all extensions E-n/E-0 are Galois
A new tower over cubic finite fields
We present a new explicit tower of function fields (Fn)nâ„0 over the finite field with ` = q3 elements, where the limit of the ratios (number of rational places of Fn)/(genus of Fn) is bigger or equal to 2(q2 â 1)/(q + 2). This tower contains as a subtower the tower which was introduced by Bezerraâ GarciaâStichtenoth (see [3]), and in the particular case q = 2 it coincides with the tower of van der Geerâvan der Vlugt (see [12]). Many features of the new tower are very similar to those of the optimal wild tower in [8] over the quadratic field Fq2 (whose modularity was shown in [6] by Elkies).
Towers of Function Fields over Non-prime Finite Fields
Over all non-prime finite fields, we construct some recursive towers of
function fields with many rational places. Thus we obtain a substantial
improvement on all known lower bounds for Ihara's quantity , for with prime and odd. We relate the explicit equations to
Drinfeld modular varieties
Further improvements on the designed minimum distance of algebraic geometry codes
In the literature about algebraic geometry codes one finds a lot of results improving Goppaâs minimum distance bound. These improvements often use the idea of âshrinkingâ or âgrowingâ the defining divisors of the codes under certain technical conditions. The main contribution of this article is to show that most of these improvements can be obtained in a unified way from one (rather simple) theorem. Our result does not only simplify previous results but it also improves them further
Asymptotics for the genus and the number of rational places in towers of function fields over a finite field
We discuss the asymptotic behaviour of the genus and the number of rational places in towers of function fields over a finite field
Galois towers over non-prime finite fields
In this paper we construct Galois towers with good asymptotic properties over
any non-prime finite field ; i.e., we construct sequences of
function fields over of increasing genus, such that all the extensions are
Galois extensions and the number of rational places of these function fields
grows linearly with the genus. The limits of the towers satisfy the same lower
bounds as the best currently known lower bounds for the Ihara constant for
non-prime finite fields. Towers with these properties are important for
applications in various fields including coding theory and cryptography
On the Value Set of n! Modulo a Prime
This is a preprint of an article published by TĂBÄ°TAK; William D. Banks, Florian Luca, Igor E. Shparlinski, Henning Stichtenoth, âOn the value set of n! modulo a prime,â Turkish Journal of Mathematics, 29 (2005), 169-174. Copyright ©2005.We show that for infinitely many prime numbers p there are at least log log p/ log log log p distinct residue classes modulo p that are not congruent to n! for any integer n
On the Invariants of Towers of Function Fields over Finite Fields
We consider a tower of function fields F=(F_n)_{n\geq 0} over a finite field
F_q and a finite extension E/F_0 such that the sequence
\mathcal{E):=(EF_n)_{n\goq 0} is a tower over the field F_q. Then we deal with
the following: What can we say about the invariants of \mathcal{E}; i.e., the
asymptotic number of places of degree r for any r\geq 1 in \mathcal{E}, if
those of F are known? We give a method based on explicit extensions for
constructing towers of function fields over F_q with finitely many prescribed
invariants being positive, and towers of function fields over F_q, for q a
square, with at least one positive invariant and certain prescribed invariants
being zero. We show the existence of recursive towers attaining the
Drinfeld-Vladut bound of order r, for any r\geq 1 with q^r a square. Moreover,
we give some examples of recursive towers with all but one invariants equal to
zero.Comment: 23 page
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