On Maximal Curves

Fuhrmann, Rainer; Garcia, Arnaldo; Torres, Fernando

doi:10.1006/jnth.1997.2148

Cited by 82 publications

(111 citation statements)

References 16 publications

(8 reference statements)

Supporting

Mentioning

107

Contrasting

Unclassified

Order By: Relevance

“…This shows that g 2 is the second largest genus for F q 2 -maximal curves. For q odd, the above curve is the only F q 2 -maximal curve (up to F q 2 -isomorphism) of genus (q 1) 2 =4 [ 7 ] . It seems plausible that uniqueness also holds true for q even but it has been so far proved under the additional Condition () below (see [1] …”

Section: On the Third Largest Genusmentioning

confidence: 99%

“…2.5, for q oddwe h a v e g 3 ( q 1)(q 2)=4 : Since X is equipped with an F q 2 -intrinsic linear series D X [7], x1, the approach due to St ohr and Voloch [36] can beapplied to investigate D X . We have dim(D X ) 2, equality holding i X is F q 2 -isomorphic to the Hermitian curve [9], Thm.…”

Section: On the Third Largest Genusmentioning

confidence: 99%

“…Here, maximality of a (projective geometrically irreducible non-singular algebraic) curve means that the number of its F-rational points attains the Hasse-Weil upper bound q 2 + 1 + 2 qg; with g being the genus of the curve and F the eld F q 2 of order q 2 . The majority of work has focused on determining the spectrum for the genera of maximal curves, and the relevant known results and open problems concern those maximal curves which h a v e large genus g with respect to the order q 2 of the underlying eld; see for example [2], [4], [7], [10], [11], [13], [14], and [15]. The precise upper bound limit is known to be q(q 1)=2 [23].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Curves of large genus covered by the hermitian curve

Cossidente¹,

Korchmáros²,

Torres

2000

Communications in Algebra

View full text Add to dashboard Cite

For the Hermitian curve H dened over the nite eld F q 2 , we give a complete classication of Galois coverings of H of prime degree. The corresponding quotient curves turn out to be special cases of wider families of curves F q 2 -covered by H arising from subgroups of the special linear group SL (2; F q ) or from subgroups in the normaliser of the Singer group of the projective unitary group P G U (3; F q 2 ). Since curves F q 2 -covered by H are maximal over F q 2 , our results provide some classication and existence theorems for maximal curves having large genus, as well as several values for the spectrum of the genera of maximal curves. For every q 2 , both the upper limit and the second largest genus in the spectrum are known, but the determination of the third largest value is still in progress. A discussion on the \third largest genus problem" including some new results and a detailed account of current w ork is given. MIRAMARE { TRIESTE

show abstract

Section: On the Third Largest Genusmentioning

confidence: 99%

Section: On the Third Largest Genusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Curves of large genus covered by the hermitian curve

Cossidente¹,

Korchmáros²,

Torres

2000

Communications in Algebra

View full text Add to dashboard Cite

show abstract

“…Theorem 4.2 (Fuhrmann-Garcia-Torres [4]). If X is a maximal curve of genus g over F q (where q is a square) then g = (q− √ q)/2 or g ≤ ( √ q−1) 2 /4.…”

Section: #X(fmentioning

confidence: 99%

Curves over finite fields with many points: an introduction

Voight

2005

Computational Aspects of Algebraic Curves

View full text Add to dashboard Cite

Abstract. The number of points on a curve defined over a finite field is bounded as a function of its genus g. In this introductory article, we survey what is known about the maximum number of points on a curve of genus g defined over Fq, including an exposition of upper bounds, lower bounds, known values of this maximum, and briefly indicate some methods of constructing curves with many points, providing many references to the literature.By the Hasse-Weil bound (also known as the Riemann hypothesis for curves over finite fields), the number of points on a smooth, geometrically integral projective curve X of genus g = g X over a finite field F q satisfiesand is therefore bounded as a function of g and q. It is natural to inquire about the sharpness of this upper bound, and so we consider the quantitywhich is the maximum number of points on a curve of genus g defined over F q , as well as the asymptotic quantityRecently, these quantities have seen a great deal of study, motivated in part by applications in coding theory and cryptography and because it is an enticing problem. This article surveys the results in this area, including an exposition of upper bounds, lower bounds, known values of N q (g), and methods of constructing curves with many points.

show abstract

“…Maximal curves have also been investigated for their applications in Coding theory. Surveys on maximal curves are found in [11,14,12,13,36,37], see also [10,9,31,35].…”

Section: Introductionmentioning

confidence: 99%