The structure of linear codes of constant weight

In this paper we determine completely the structure of linear codes over of constant weight. Namely, we determine exactly which modules underlie linear codes of constant weight, and we describe the coordinate functionals involved. The weight functions considered are: Hamming weight, Lee weight, two forms of Euclidean weight, and pre-homogeneous weights. We prove a general uniqueness theorem for virtual linear codes of constant weight. Existence is settled on a case by case basis.

     

Focal loci of families and the genus of curves on surfaces

In this article we apply the classical method of focal loci of families to give a lower bound for the genus of curves lying on general surfaces. First we translate and reprove Xu's result that any curve on a general surface in of degree has geometric genus . Then we prove a similar lower bound for the curves lying on a general surface in a given component of the Noether-Lefschetz locus in and on a general projectively Cohen-Macaulay surface in .

     

Examples of genus two CM curves defined over the rationals

We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the well-known example we find 19 non-isomorphic such curves. We believe that these are the only such curves.

     

Toeplitz operators with PC symbols on general Carleson Jordan curves with arbitrary Muckenhoupt weights

We describe the spectra and essential spectra of Toeplitz operators with piecewise continuous symbols on the Hardy space in case , is a Carleson Jordan curve and is a Muckenhoupt weight in . Classical results tell us that the essential spectrum of the operator is obtained from the essential range of the symbol by filling in line segments or circular arcs between the endpoints of the jumps if both the curve and the weight are sufficiently nice. Only recently it was discovered by Spitkovsky that these line segments or circular arcs metamorphose into horns if the curve is nice and is an arbitrary Muckenhoupt weight, while the authors observed that certain special so-called logarithmic leaves emerge in the case of arbitrary Carleson curves with nice weights. In this paper we show that for general Carleson curves and general Muckenhoupt weights the sets in question are logarithmic leaves with a halo, and we present final results concerning the shape of the halo.

     

Regular components of moduli spaces of stable maps

We construct regular components of the moduli space of stable maps from curves of genus to a product of two projective spaces. These components are generically smooth and have the expected dimension predicted by deformation theory. This result can be seen as a general position theorem for loci in consisting of curves carrying exceptional linear series.

     

Heegner divisors in the moduli space of genus three curves

S. Kondo used periods of surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6-ball. In this paper we study Heegner divisors in the ball quotient, given by arithmetically defined hyperplane sections of the ball. We show that the corresponding loci of genus three curves are given by hyperelliptic curves, singular plane quartics and plane quartics admitting certain rational ``splitting curves'.

     

A flexible method for applying Chabauty's Theorem

A strategy is proposed for applying Chabauty's Theoremto hyperelliptic curves of genus > 1. In the genus 2case, it shown how recent developments on the formal group of the Jacobiancan be used to give a flexible and computationallyviable method for applying this strategy. The details are described for a general curveof genus 2, and are then applied to find C() fora selection of curves. A fringe benefit is a more explicit proof of a result of Coleman.     

A Gap in GRM Code Weight Distributions

The weight distribution of GRM (generalized Reed-Muller) codes is unknown in general. This article describes and applies some new techniques to the codes over F3. Specifically, we decompose GRM codewords into words from smaller codes and use this decomposition, along with a projective geometry technique, to relate weights occurring in one code with weights occurring in simpler codes. In doing so, we discover a new gap in the weight distribution of many codes. In particular, we show there is no word of weight 3^m–2 in GRM₃(4,m) for m>6, and for even-order codes over the ternary field, we show that under certain conditions, there is no word of weight d+, where d is the minimum distance and is the largest integer dividing all weights occurring in the code.     

On a unit group generated by special values of Siegel modular functions

There has been important progress in constructing units and -units associated to curves of genus 2 or 3. These approaches are based mainly on the consideration of properties of Jacobian varieties associated to hyperelliptic curves of genus 2 or 3. In this paper, we construct a unit group of the ray class field of modulo 6 with full rank by special values of Siegel modular functions and circular units. We note that . Our construction of units is number theoretic, and closely based on Shimura's work describing explicitly the Galois actions on the special values of theta functions.

     

On the classification of irregular surfaces of general type with nonbirational bicanonical map

The present paper is devoted to the classification of irregular surfaces of general type with and nonbirational bicanonical map. Our main result is that, if is such a surface and if is minimal with no pencil of curves of genus , then is the symmetric product of a curve of genus , and therefore and . Furthermore we obtain some results towards the classification of minimal surfaces with . Such surfaces have , and we show that if and only if is the symmetric product of a curve of genus . We also classify the minimal surfaces with with a pencil of curves of genus , proving in particular that for those one has .

     

A Class of Perfect Ternary Constant-Weight Codes

We construct a class of perfect ternary constant-weight codes of length 2 ^r , weight 2 ^r -1 and minimum distance 3. The codes have codewords. The construction is based on combining cosets of binary Hamming codes. As a special case, for r=2 the construction gives the subcode of the tetracode consisting of its nonzero codewords. By shortening the perfect codes, we get further optimal codes.     

Distinguishing embedded curves in rational complex surfaces

We construct many pairs of smoothly embedded complex curves with the same genus and self-intersection number in the rational complex surfaces with the property that no self-diffeomorphism of sends one to the other. In particular, as a special case we answer a question originally posed by R. Gompf (1995) concerning genus two curves of self-intersection number 0 in .

     

On Drinfeld Modular Curves with Many Rational Points over Finite Fields

It is known that Drinfeld modular curves can be used to construct asymptotically optimal towers of curves over finite fields. Using reductions of the Drinfeld modular curves X₀( ), we try to find individual curves over finite fields with many rational points. The main idea is to divide by an Atkin–Lehner involution which has many fixed points in order to obtain a quotient with a better ratio #{rational points}/genus. In a few cases we can improve the known records of rational points.     

On Linear Codes over $${\mathbb{Z}}_{2^{s}}$$

In an earlier paper the authors studied simplex codes of type α and β over and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over The above codes are also shown to satisfy the chain condition.A part of this paper is contained in his Ph.D. Thesis from IIT Kanpur, India     

Integral points on elliptic curves and -torsion in class groups

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques and methods based on quasi-orthogonality in the Mordell-Weil lattice. We apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the -torsion part of the class group of a quadratic field. The same ideas can be used to count rational points on curves of higher genus.

     

Correlated algebraic-geometric codes: Improved list decoding over bounded alphabets

We define a new family of error-correcting codes based on algebraic curves over finite fields, and develop efficient list decoding algorithms for them. Our codes extend the class of algebraic-geometric (AG) codes via a (nonobvious) generalization of the approach in the recent breakthrough work of Parvaresh and Vardy (2005).

Our work shows that the PV framework applies to fairly general settings by elucidating the key algebraic concepts underlying it. Also, more importantly, AG codes of arbitrary block length exist over fixed alphabets , thus enabling us to establish new trade-offs between the list decoding radius and rate over a bounded alphabet size.

The work of Parvaresh and Vardy (2005) was extended in Guruswami and Rudra (2006) to give explicit codes that achieve the list decoding capacity (optimal trade-off between rate and fraction of errors corrected) over large alphabets. A similar extension of this work along the lines of Guruswami and Rudra could have substantial impact. Indeed, it could give better trade-offs than currently known over a fixed alphabet (say, ), which in turn, upon concatenation with a fixed, well-understood binary code, could take us closer to the list decoding capacity for binary codes. This may also be a promising way to address the significant complexity drawback of the result of Guruswami and Rudra, and to enable approaching capacity with bounded list size independent of the block length (the list size and decoding complexity in their work are both where is the distance to capacity).

Similar to algorithms for AG codes from Guruswami and Sudan (1999) and (2001), our encoding/decoding algorithms run in polynomial time assuming a natural polynomial-size representation of the code. For codes based on a specific ``optimal' algebraic curve, we also present an expected polynomial time algorithm to construct the requisite representation. This in turn fills an important void in the literature by presenting an efficient construction of the representation often assumed in the list decoding algorithms for AG codes.

     

Embedded surfaces and almost complex structures

In this paper, we prove necessary and sufficient conditions for a smooth surface in a smooth 4-manifold to be pseudoholomorphic with respect to an almost complex structure on . In particular, this provides a systematic approach to the construction of pseudoholomorphic curves that do not minimize the genus in their homology class.

     

Real forms of a Riemann surface of even genus

Natanzon proved that a Riemann surface of genus has at most conjugacy classes of symmetries, and this bound is attained for infinitely many genera . The aim of this note is to prove that a Riemann surface of even genus has at most four conjugacy classes of symmetries and this bound is attained for an arbitrary even as well. An equivalent formulation in terms of algebraic curves is that a complex curve of an even genus has at most four real forms which are not birationally equivalent.