ON INVARIANTS OF TOWERS OF FUNCTION FIELDS OVER FINITE FIELDS

In this paper, we consider a tower of function fields [Formula: see text] over a finite field 𝔽q and a finite extension E/F0 such that the sequence [Formula: see text] is a tower over the field 𝔽q. Then we study invariants of [Formula: see text], that is, the asymptotic number of the places of degree r in [Formula: see text], for any r ≥ 1, if those of [Formula: see text] are known. We first give a method for constructing towers of function fields over any finite field 𝔽q with finitely many prescribed invariants being positive. For q a square, we prove that with the same method one can also construct towers with at least one positive invariant and certain prescribed invariants being zero. Our method is based on explicit extensions. Moreover, we show the existence of towers over a finite field 𝔽q attaining the Drinfeld–Vladut bound of order r, for any r ≥ 1 with qr a square (see [1, Problem-2]). Finally, we give some examples of non-optimal recursive towers with all but one invariants equal to zero.

Download Full-text

A link between minimal value set polynomials and tamely ramified towers of function fields over finite fields

Journal of Algebra and Its Applications ◽

10.1142/s0219498822501894 ◽

2021 ◽

pp. 2250189

Author(s):

R. Toledano

Keyword(s):

Finite Field ◽

Finite Fields ◽

Finite Subset ◽

Function Fields ◽

Algebraic Closure ◽

Value Set ◽

Towers Of Function Fields ◽

Value Sets

In this paper, we introduce the notions of [Formula: see text]-polynomial and [Formula: see text]-minimal value set polynomial where [Formula: see text] is a polynomial over a finite field [Formula: see text] and [Formula: see text] is a finite subset of an algebraic closure of [Formula: see text]. We study some properties of these polynomials and we prove that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are [Formula: see text]-minimal value set polynomials for [Formula: see text], whose [Formula: see text]-value sets can be explicitly computed in terms of the monomial [Formula: see text].

Download Full-text

Towers of function fields over finite fields corresponding to elliptic modular curves

Finite Fields and Their Applications ◽

10.1016/j.ffa.2011.04.003 ◽

2012 ◽

Vol 18 (1) ◽

pp. 1-18 ◽

Cited By ~ 1

Author(s):

Takehiro Hasegawa ◽

Miyoko Inuzuka ◽

Takafumi Suzuki

Keyword(s):

Finite Fields ◽

Function Fields ◽

Modular Curves ◽

Towers Of Function Fields

Download Full-text

New Optimal Tame Towers of Function Fields over Small Finite Fields

Lecture Notes in Computer Science - Algorithmic Number Theory ◽

10.1007/3-540-45455-1_30 ◽

2002 ◽

pp. 372-389 ◽

Cited By ~ 2

Author(s):

Wen-Ching W. Li ◽

Hiren Maharaj ◽

Henning Stichtenoth ◽

Noam D. Elkies

Keyword(s):

Finite Fields ◽

Function Fields ◽

Towers Of Function Fields

Download Full-text

A note on towers of function fields over finite fields

Communications in Algebra ◽

10.1080/00927879808826370 ◽

1998 ◽

Vol 26 (11) ◽

pp. 3737-3741 ◽

Cited By ~ 1

Author(s):

Ferruh Özbudak ◽

Michael Thomas

Keyword(s):

Finite Fields ◽

Function Fields ◽

Towers Of Function Fields

Download Full-text

Asymptotics for the genus and the number of rational places in towers of function fields over a finite field

Finite Fields and Their Applications ◽

10.1016/j.ffa.2005.04.004 ◽

2005 ◽

Vol 11 (3) ◽

pp. 434-450 ◽

Cited By ~ 2

Author(s):

Arnaldo Garcia ◽

Henning Stichtenoth

Keyword(s):

Finite Field ◽

Function Fields ◽

Rational Places ◽

Towers Of Function Fields

Download Full-text

Recursive Towers of Function Fields over Finite Fields

Arithmetic of Finite Fields - Lecture Notes in Computer Science ◽

10.1007/978-3-642-13797-6_1 ◽

2010 ◽

pp. 1-6

Author(s):

Henning Stichtenoth

Keyword(s):

Finite Fields ◽

Function Fields ◽

Towers Of Function Fields

Download Full-text

Diagonal Equations Over Large Finite Fields

Canadian Journal of Mathematics ◽

10.4153/cjm-1984-016-6 ◽

1984 ◽

Vol 36 (2) ◽

pp. 249-262 ◽

Cited By ~ 2

Author(s):

Charles Small

Keyword(s):

Finite Field ◽

Finite Fields ◽

Finite Extension ◽

Extension Field ◽

Explicit Bound ◽

Image Position

We consider polynomials of the formwith non-zero coefficients ai in a finite field F. For any finite extension field K ⊇ F, let fk:Kn → K be the mapping defined by f. We say f is universal over K if fK is surjective, and f is isotropic over K if fK has a non-trivial “kernel“; the latter means fK(X) = 0 for some 0 ≠ x ∊ Kn.We show (Theorem 1) that f is universal over K provided |K| (the cardinality of K) is larger than a certain explicit bound given in terms of the exponents d1,…, dn. The analogous fact for isotropy is Theorem 2.It should be noted that in studying diagonal equationswe fix both the number of variables n and the exponents di, and ask how large the field must be to guarantee a solution.

Download Full-text