Tokyo Journal of Mathematics

A Note on Optimal Towers over Finite Fields


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Recently, under the influence of Elkies' conjecture [2], the optimal recursive towers of algebraic function fields (of one variable) over the finite fields with square cardinality are studied [3, 5, 6]. In this paper, we define the limit $$ \lim_{i \to \infty} \text{(the number of places of degree $n$ in $F_{i}/\mathbf{F}_{q}$)/(genus of $F_{i}$)} $$ of a tower $F_{0} \subseteq F_{1} \subseteq F_{2} \subseteq \cdots$ over the finite field $\F_{q}$. Using this limit, we prove that all the proper constant field extensions of all the optimal towers over the finite fields with square cardinality are not optimal, and we show a simple criterion whether a tower is optimal or not. Moreover, we give many new recursive towers of finite ramification type.

Article information

Tokyo J. Math., Volume 30, Number 2 (2007), 477-487.

First available in Project Euclid: 4 February 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R58: Arithmetic theory of algebraic function fields [See also 14-XX]
Secondary: 14G15: Finite ground fields 14H05: Algebraic functions; function fields [See also 11R58]


HASEGAWA, Takehiro. A Note on Optimal Towers over Finite Fields. Tokyo J. Math. 30 (2007), no. 2, 477--487. doi:10.3836/tjm/1202136690.

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  • V. G. Drinfeld and S. G. Vlăduţ, Number of points of an algebraic curve, Functional Anal. Appl., 17 (1983), 53–54.
  • N. D. Elkies, Explicit modular towers, in: T. Basar and A. Vardy (eds.), Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control and Computing (1997), 23–32.
  • A. Garcia and H. Stichtenoth, On the asymptotic behavior of some towers of function fields over finite fields, J. Number Theory, 61 (1996), 248–273.
  • A. Garcia and H. Stichtenoth, Skew pyramids of function fields are asymptotically bad, in: J. Buchmann et al. (eds.), Coding Theory, Cryptography and Related Areas, Springer-Verlag Berlin, 2000, 111–113.
  • A. Garcia, H. Stichtenoth and H. G. Rück, On tame towers over finite fields, J. Reine Angew. Math., 557 (2003), 53–80.
  • A. Garcia, H. Stichtenoth and M. Thomas, On towers and composita of towers of function fields over finite fields, Finite Fields Appl., 3 (1997), 257–274.
  • T. Hasegawa, An upper bound for the Garcia-Stichtenoth numbers of towers, Tokyo J. Math., 28 (2005), 471–481.
  • Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA. Math., 28 (1981), 721–724.
  • H. Stichtenoth, Algebraic function fields and codes, Springer-Verlag Berlin, 1993.
  • M. A. Tsfasman, Some remarks on the asymptotic number of points, in: H. Stichtenoth and M. A. Tsfasman (Eds.), Coding theory and algebraic geometry, Springer-Verlag Berlin, 1992, 178–192.
  • M. A. Tsfasman and S. G. Vlăduţ, Algebraic-geometric codes, Kluwer Dordrecht, 1991.
  • M. A. Tsfasman, S. G. Vlăduţ and T. Zink, Modular curves, Shimura curves and Goppa codes, better than the Varshamov-Gilbert bound, Math. Nachr., 109 (1982), 21–28.