An explicit tower of function fields over cubic finite fields and Zink’s lower bound

2005 ◽  
Vol 2005 (589) ◽  
pp. 159-199 ◽  
Author(s):  
Juscelino Bezerra ◽  
Arnaldo Garcia ◽  
Henning Stichtenoth
Keyword(s):  
Lower Bound ◽  
Finite Fields ◽  
Function Fields ◽  
Tower Of Function Fields

scholarly journals ON INVARIANTS OF TOWERS OF FUNCTION FIELDS OVER FINITE FIELDS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250190 ◽  
Author(s):  
FLORIAN HESS ◽  
HENNING STICHTENOTH ◽  
SEHER TUTDERE
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Function Fields ◽  
Finite Extension ◽  
Towers Of Function Fields ◽  
Asymptotic Number ◽  
Tower Of Function 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.

scholarly journals Extremal behaviour of ± 1-valued completely multiplicative functions in function fields

2021 ◽  
Vol 56 (1) ◽  
pp. 79-94
Author(s):  
Nikola Lelas ◽  
Keyword(s):  
Rational Function ◽  
Finite Fields ◽  
Function Fields ◽  
Mean Value ◽  
Multiplicative Functions ◽  
Irreducible Polynomials ◽  
Liouville Function ◽  
The Mean

We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].

ON THE -ERROR LINEAR COMPLEXITY OF SEQUENCES FROM FUNCTION FIELDS

2020 ◽  
Vol 102 (2) ◽  
pp. 342-352
Author(s):  
YUHUI ZHOU ◽  
YUHUI HAN ◽  
YANG DING
Keyword(s):  
Lower Bound ◽  
Linear Complexity ◽  
Function Fields ◽  
Stream Ciphers ◽  
Security Measures ◽  
Periodic Sequences ◽  
Error Linear Complexity

The linear complexity and the error linear complexity are two important security measures for stream ciphers. We construct periodic sequences from function fields and show that the error linear complexity of these periodic sequences is large. We also give a lower bound for the error linear complexity of a class of nonperiodic sequences.

A Lower Bound on the Number of Cyclic Function Fields With Class Number Divisible byn

2006 ◽  
Vol 49 (3) ◽  
pp. 448-463 ◽  
Author(s):  
Allison M. Pacelli
Keyword(s):  
Lower Bound ◽  
Class Number ◽  
Function Fields ◽  
Quadratic Function ◽  
Class Numbers ◽  
Prime Degree ◽  
Cyclic Function ◽  
Quadratic Function Fields

AbstractIn this paper, we find a lower bound on the number of cyclic function fields of prime degreelwhose class numbers are divisible by a given integern. This generalizes a previous result of D. Cardon and R. Murty which gives a lower bound on the number of quadratic function fields with class numbers divisible byn.

New Optimal Tame Towers of Function Fields over Small Finite Fields

2002 ◽  
pp. 372-389 ◽  
Author(s):  
Wen-Ching W. Li ◽  
Hiren Maharaj ◽  
Henning Stichtenoth ◽  
Noam D. Elkies
Keyword(s):  
Finite Fields ◽  
Function Fields ◽  
Towers Of Function Fields

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

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