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Mosc. Math. J., 2015, Volume 15, Number 1, Pages 1–29 (Mi mmj546)  

This article is cited in 16 scientific papers (total in 16 papers)

Towers of function fields over non-prime finite fields

Alp Bassaa, Peter Beelenb, Arnaldo Garciac, Henning Stichtenotha

a Sabancı University, MDBF, 34956 Tuzla, İstanbul, Turkey
b Technical University of Denmark, Department of Applied Mathematics and Computer Science, Matematiktorvet, Building 303B, DK-2800, Lyngby, Denmark
c Instituto Nacional de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, RJ, Brazil

Abstract: 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 $A(\ell)$, for $\ell=p^n$ with $p$ prime and $n>3$ odd. We relate the explicit equations to Drinfeld modular varieties.

Key words and phrases: curves with many points, towers of function fields, genus, rational places, Ihara's constant.


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MSC: 11G20, 11G09, 11R58
Received: July 2, 2014

Citation: Alp Bassa, Peter Beelen, Arnaldo Garcia, Henning Stichtenoth, “Towers of function fields over non-prime finite fields”, Mosc. Math. J., 15:1 (2015), 1–29

Citation in format AMSBIB
\by Alp~Bassa, Peter~Beelen, Arnaldo~Garcia, Henning~Stichtenoth
\paper Towers of function fields over non-prime finite fields
\jour Mosc. Math.~J.
\yr 2015
\vol 15
\issue 1
\pages 1--29

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    This publication is cited in the following articles:
    1. H. Stichtenoth, S. Tutdere, “Quadratic recursive towers of function fields over $\mathbb F_2$”, Turkish J. Math., 39:5 (2015), 665–682  crossref  mathscinet  zmath  isi  scopus
    2. G. Cohen, S. Mesnager, H. Randriam, “Yet another variation on minimal linear codes”, Adv. Math. Commun., 10:1 (2016), 53–61  crossref  mathscinet  zmath  isi  scopus
    3. E. Hallouin, M. Perret, “A graph aided strategy to produce good recursive towers over finite fields”, Finite Fields Appl., 42 (2016), 200–224  crossref  mathscinet  zmath  isi  scopus
    4. N. Anbar, A. Bassa, P. Beelen, “A modular interpretation of various cubic towers”, J. Number Theory, 171 (2017), 341–357  crossref  mathscinet  zmath  isi  scopus
    5. O. Geil, S. Martin, U. Martinez-Penas, R. Matsumoto, D. Ruano, “On asymptotically good ramp secret sharing schemes”, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., E100A:12 (2017), 2699–2708  crossref  isi  scopus
    6. Ch. Hu, “Explicit construction of AG codes from a curve in the tower of Bassa-Beelen-Garcia-Stichtenoth”, IEEE Trans. Inf. Theory, 63:11 (2017), 7237–7246  crossref  mathscinet  zmath  isi  scopus
    7. N. Anbar, P. Beelen, “A note on a tower by Bassa, Garcia and Stichtenoth”, Funct. Approx. Comment. Math., 57:1 (2017), 47–60  crossref  mathscinet  zmath  isi
    8. L. You, F. Knoll, Yu. Mao, Sh. Gao, “Practical Johnson–Lindenstrauss transforms via algebraic geometry codes”, 2017 International Conference on Control, Artificial Intelligence, Robotics & Optimization (Iccairo), IEEE, 2017, 171–176  crossref  isi  scopus
    9. N. Anbar, P. Beelen, Nhut Nguyen, “The exact limit of some cubic towers”, Arithmetic, Geometry, Cryptography and Coding Theory, Contemporary Mathematics, 686, eds. A. Bassa, A. Couvreur, D. Kohel, Amer. Math. Soc., 2017, 1–15  crossref  mathscinet  zmath  isi  scopus
    10. N. Anbar, P. Beelen, Nhut Nguyen, “A new tower with good $p$-rank meeting Zink's bound”, Acta Arith., 177:4 (2017), 347–374  crossref  mathscinet  zmath  isi  scopus
    11. S. G. Vlăduţ, D. Yu. Nogin, M. A. Tsfasman, “Varieties over finite fields: quantitative theory”, Russian Math. Surveys, 73:2 (2018), 261–322  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. L. Jin, Ch. Xing, “Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound”, IEEE Trans. Inf. Theory, 64:9 (2018), 6277–6282  crossref  mathscinet  zmath  isi  scopus
    13. R. Pellikaan, X. W. Wu, S. Bulygin, R. Jurrius, Codes, Cryptology and Curves With Computer Algebra, Cambridge Univ. Press, 2018  crossref  mathscinet  zmath  isi  scopus
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    15. E.-U. Gekeler, “Towers of gl(r)-type of modular curves”, J. Reine Angew. Math., 754 (2019), 87–141  crossref  mathscinet  zmath  isi  scopus
    16. J. Vergara-Morales, M. del Valle Tapia, A. Diaz Mujica, L. Matos Fernandez, M. V. Perez Villalobos, “Mediating effect of autonomous motivation in learning”, Rev. Electron. Invest. Educ., 21 (2019), UNSP e37  crossref  isi  scopus
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