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Recently, two-to-one mappings have received a lot of attention due to their applications to cryptography and to the construction of classes of cryptographic functions. In this paper, we propose a new method to obtain m -to-1 functions based on Galois groups of rational functions. We feel that such an approach might be the starting point of a new line of research in the area of m -to-1 mappings. [ABSTRACT FROM AUTHOR]

Let X be a (projective, geometrically irreducible, non-singular) algebraic curve of genus g ≥ 2 defined over an algebraically closed field K of odd characteristic p. Let Aut (X) be the group of all automorphisms of X which fix K element-wise. It is known that if | Aut (X) | ≥ 8 g 3 then the p -rank (equivalently, the Hasse–Witt invariant) of X is zero. This raises the problem of determining the (minimum-value) function f (g) such that whenever | Aut (X) | ≥ f (g) then X has zero p -rank. For even g we prove that f (g) ≤ 900 g 2. The odd genus case appears to be much more difficult although, for any genus g ≥ 2 , if Aut (X) has a solvable subgroup G such that | G | > 252 g 2 then X has zero p -rank and G fixes a point of X. Our proofs use the Hurwitz genus formula and the Deuring Shafarevich formula together with a few deep results from Group theory characterizing finite simple groups whose Sylow 2-subgroups have a cyclic subgroup of index 2. We also point out some connections with the Abhyankar conjecture and the Katz–Gabber covers. [ABSTRACT FROM AUTHOR]

Abstract The Dickson–Guralnick–Zieve curve, briefly DGZ curve, defined over the finite field F q arises naturally from the classical Dickson invariant of the projective linear group P G L (3 , F q). The DGZ curve is an (absolutely irreducible, singular) plane curve of degree q 3 − q 2 and genus 1 2 q (q − 1) (q 3 − 2 q − 2) + 1. In this paper we show that the DGZ curve has several remarkable features, those appearing most interesting are: the DGZ curve has a large automorphism group compared to its genus albeit its Hasse–Witt invariant is positive; the Fermat curve of degree q − 1 is a quotient curve of the DGZ curve; among the plane curves with the same degree and genus of the DGZ curve and defined over F q 3 , the DGZ curve is optimal with respect the number of its F q 3 -rational points. [ABSTRACT FROM AUTHOR]

In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over F 3 2 k . In addition, new examples and generalizations of some families of permutation polynomials of F 3 k and F 5 k are given. We also study permutation quadrinomials of type A x q ( q − 1 ) + 1 + B x 2 ( q − 1 ) + 1 + C x q + x . Our method is based on the investigation of an algebraic curve associated with a fractional polynomial over a finite field. [ABSTRACT FROM AUTHOR]

Let S be a p -subgroup of the K -automorphism group Aut ( X ) of an algebraic curve X of genus g ≥ 2 and p -rank γ defined over an algebraically closed field K of characteristic p ≥ 3 . Nakajima [27] proved that if γ ≥ 2 then | S | ≤ p p − 2 ( g − 1 ) . If equality holds, X is a Nakajima extremal curve . We prove that if | S | > p 2 p 2 − p − 1 ( g − 1 ) then one of the following cases occurs. (i) γ = 0 and the extension K ( X ) | K ( X ) S completely ramifies at a unique place, and does not ramify elsewhere. (ii) | S | = p , and X is an ordinary curve of genus g = p − 1 . (iii) X is an ordinary, Nakajima extremal curve, and K ( X ) is an unramified Galois extension of a function field of a curve given in (ii). (iii) X is an ordinary, Nakajima extremal curve, and K ( X ) is an unramified Galois extension of a function field of a curve given in (ii). There are exactly p − 1 subgroups M of S such that K ( X ) | K ( X ) M is such a Galois extension. Moreover, if some of them is an abelian extension then S has maximal nilpotency class. The full K -automorphism group of any Nakajima extremal curve is determined, and several infinite families of Nakajima extremal curves are constructed by using their pro- p fundamental groups. [ABSTRACT FROM AUTHOR]

Let S be a 2-subgroup of the K -automorphism group Aut ( X ) of an algebraic curve X of genus g ( X ) defined over an algebraically closed field K of characteristic 2. It is known that S may be quite large compared to the classical Hurwitz bound 84 ( g ( X ) − 1 ) . However, if S fixes no point, then the size of S is smaller than or equal to 4 ( g ( X ) − 1 ) . In this paper, we investigate algebraic curves X with a 2-subgroup S of Aut ( X ) having the following properties: (I) | S | ≥ 8 and | S | > 2 ( g ( X ) − 1 ) , (II) S fixes no point on X . Theorem 1.2 shows that X is a general curve and that either | S | = 4 ( g ( X ) − 1 ) , or | S | = 2 g ( X ) + 2 , or, for every involution u ∈ Z ( S ) , the quotient curve X / 〈 u 〉 inherits the above properties, that is, it has genus ≥2, and its automorphism group S / 〈 u 〉 still has properties (I) and (II). In the first two cases, S is completely determined. We also give examples illustrating our results. In particular, for every g = 2 h + 1 ≥ 9 , we exhibit a (general bielliptic) curve X of genus g whose K -automorphism group has a dihedral 2-subgroup S of order 4 ( g − 1 ) that fixes no point in X . [ABSTRACT FROM AUTHOR]

The study of k-arcs in AG(2, q) determining few directions is motivated by applications to secret sharing schemes. In this paper we obtain some new results for odd q. An essential tool comes from algebraic geometry via Segre's result on algebraic envelopes associated to a k-arc. [ABSTRACT FROM AUTHOR]

Abstract: We fill up a gap in Hennʼs proof concerning large automorphism groups of function fields of degree 1 over an algebraically closed field of positive characteristic. [Copyright &y& Elsevier]

The Stöhr-Voloch approach has been largely used to deal with the classical problem of estimating the number of rational points of a Fermat curve over a finite field. The same method actually applies to any curve admitting as an automorphism group the direct product of two cyclic groups C1 and C2 of the same size k, and such that the quotient curves with respect to both C1 and C2 are rational. In this paper such a curve is called a generalized Fermat curve. Our main achievement is that of extending some known results on Fermat curves to generalized Fermat curves. [ABSTRACT FROM AUTHOR]

The Hurwitz bound on the size of the -automorphism group Aut() of an algebraic curve  of genus g ≥ 2 defined over a field  of zero characteristic is |Aut()| ≤ 84(g − 1). For a positive characteristic, algebraic curves can have many more automorphisms than expected from the Hurwitz bound. There even exist algebraic curves of arbitrary high genus g with more than 16g4 automorphisms. It has been observed on many occasions that the most anomalous examples of algebraic curves with very large automorphism groups invariably have zero p-rank. In this paper, the -automorphism group Aut() of a zero 2-rank algebraic curve  defined over an algebraically closed field  of characteristic 2 is investigated. The main result is that, if the curve has genus g ≥ 2 and |Aut()| > 24g(g − 1), then Aut() has a fixed point on , apart from a few exceptions. In the exceptional cases, the possibilities for Aut() and g are determined. [ABSTRACT FROM PUBLISHER]

In this paper, algebraic-geometric (AG) codes associated to a recently discovered class of maximal curves are investigated. As a result, some linear codes with better parameters with respect to the previously known ones are discovered, and 70 improvements on MinT's tables ("Tables of optimal parameters for linear codes," University of Salzburg, Salzburg, Austria) are obtained. [ABSTRACT FROM AUTHOR]

Abstract: Complete n-tracks in and non-extendable Near MDS codes of dimension over are known to be equivalent objects. The best known lower bound for the maximum number of points of an n-track is attained by elliptic n-tracks, that is, n-tracks consisting of the -rational points of an elliptic curve. This has given a motivation for the study of complete elliptic n-tracks. From previous work, an elliptic n-track in is complete provided that either the j-invariant of the underlying elliptic curve is different from zero, or and the number of -rational points of is even. In this paper it is shown that the latter result extends to odd if and only if either q is a square or , p being the characteristic of . Some completeness results for elliptic n-tracks in dimensions 3 and 5 are also obtained. [Copyright &y& Elsevier]

In this paper, explicit equations for algebraic curves with genus 4, 5, and 10 already studied in characteristic zero, are analyzed in positive characteristic p. We show that these curves have an interesting behaviour on the number of their rational places. Namely, they are either maximal or minimal over the finite field with p 2 elements for infinitely many p 's. The key tool is the investigation of their Jacobian decomposition. Lists of small p 's for which maximality holds are provided. In some cases we also describe the automorphism group of the curve. [ABSTRACT FROM AUTHOR]