The dual geometry of Hermitian two-point codes

In this paper, we study the algebraic geometry of any two-point code on the Hermitian curve and reveal the purely geometric nature of their dual minimum distance. We describe the minimum-weight codewords of many of their dual codes through an explicit geometric characterization of their supports. In particular, we show that they appear as sets of collinear points in many cases.     

LDPC codes from the Hermitian curve

In this paper, we study the code which has as parity check matrix the incidence matrix of the design of the Hermitian curve and its (q + 1)-secants. This code is known to have good performance with an iterative decoding algorithm, as shown by Johnson and Weller in (Proceedings at the ICEE Globe com conference, Sanfrancisco, CA, 2003). We shall prove that has a double cyclic structure and that by shortening in a suitable way it is possible to obtain new codes which have higher code-rate. We shall also present a simple way to constructing the matrix via a geometric approach.      

Topology of complete intersections

Let X _n (d) and X _n (d') be two n-dimensional complete intersections with the same total degree d. In this paper we prove that, if n is even and d has no prime factors less than , then X _n (d) and X _n (d') are homotopy equivalent if and only if they have the same Euler characteristics and signatures. This confirms a conjecture of Libgober and Wood [16]. Furthermore, we prove that, if d has no prime factors less than , then X _n (d) and X _n (d') are homeomorphic if and only if their Pontryagin classes and Euler characteristics agree. Received: September 6, 1996     

Structure of functional codes defined on non-degenerate Hermitian varieties

We study the functional codes of order h defined by G. Lachaud on a non-degenerate Hermitian variety, by exhibiting a result on divisibility for all the weights of such codes. In the case where the functional code is defined by evaluating quadratic functions on the non-degenerate Hermitian surface, we list the first five weights, describe the geometrical structure of the corresponding quadrics and give a positive answer to a conjecture formulated on this question by Edoukou (2009) [8]. The paper ends with two conjectures. The first is about the divisibility of the weights in the functional codes. The second is about the minimum distance and the distribution of the codewords of the first 2h+1 weights.     

Small weight codewords in the LDPC codes arising from linear representations of geometries

In this article, we investigate the minimum distance and small weight codewords of the LDPC codes of linear representations, using only geometrical methods. First, we present a new lower bound on the minimum distance and we present a number of cases in which this lower bound is sharp. Then we take a closer look at the cases and with a hyperoval, hence q even, and characterize codewords of small weight. When investigating the small weight codewords of , we deal with the case of a regular hyperoval, that is, a conic and its nucleus, separately, since in this case, we have a larger upper bound on the weight for which the results are valid. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 1–24, 2009     

Cayley-Bacharach and evaluation codes on complete intersections

Hansen (Appl. Algebra Eng. Comm. Comput. 14 (2003) 175) uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in P². In this paper, we generalize Hansen's results from P² to P^m; we also show that the hypotheses of Hansen (2003) may be weakened. The proof is succinct and follows by combining the Cayley-Bacharach Theorem and the bounds on evaluation codes obtained in Hansen (Zero-Dimensional Schemes (Ravello, 1992), de Gruyter, Berlin, 1994, pp. 205-211).     

A construction of Gray codes inducing complete graphs

A binary Gray code G(n) of length n, is a list of all 2ⁿn-bit codewords such that successive codewords differ in only one bit position. The sequence of bit positions where the single change occurs when going to the next codeword in G(n), denoted by S(n)?s₁,s₂,…,s_2ⁿ-1, is called the transition sequence of the Gray code G(n). The graph G_G(n) induced by a Gray code G(n) has vertex set {1,2,…,n} and edge set {{s_i,s_i+1}:1?i?2ⁿ-2}. If the first and the last codeword differ only in position s_2ⁿ, the code is cyclic and we extend the graph by two more edges {s_2ⁿ-1,s_2ⁿ} and {s_2ⁿ,s₁}. We solve a problem of Wilmer and Ernst [Graphs induced by Gray codes, Discrete Math. 257 (2002) 585-598] about a construction of an n-bit Gray code inducing the complete graph K_n. The technique used to solve this problem is based on a Gray code construction due to Bakos [A. Ádám, Truth Functions and the Problem of their Realization by Two-Terminal Graphs, Akadémiai Kiadó, Budapest, 1968], and which is presented in D.E. Knuth [The Art of Computer Programming, vol. 4, Addison-Wesley as part of “fascicle” 2, USA, 2005].     

Minimal degree liftings in characteristic 2

In this paper we analyze liftings of hyperelliptic curves over perfect fields in characteristic 2 to curves over rings of Witt vectors. This theory can be applied to construct error-correcting codes; lifts of points with minimal degrees are likely to yield the best codes, and these are the main focus of the paper. We find upper and lower bounds for their degrees, give conditions to achieve the lower bounds and analyze the existence of lifts of the Frobenius. Finally, we exhibit explicit computations for genus 2 and show codes obtained using this theory.     

On modules of finite projective dimension over complete intersections

Recently Avramov and Miller proved that over a local complete intersection ring in characteristic 0$">, a finitely generated module has finite projective dimension if for some 0$"> and for some 0$">, being the frobenius map repeated times. They used the notion of ``complexity' and several related theorems. Here we offer a very simple proof of the above theorem without using ``complexity' at all.

     

On Certain Subcovers of the Hermitian Curve

We present a simple construction that gives explicit equations for certain subcovers of the Hermitian curve. We show that maximal curves with a certain type of defining equations are covered by the Hermitian curve.     

The second generalized Hamming weight for two-point codes on a Hermitian curve

The aim of this article is the determination of the second generalized Hamming weight of any two-point code on a Hermitian curve of degree q + 1. The determination involves results of Coppens on base-point-free pencils on a plane curve. To avoid non- essential trouble, we assume that q > 4.      

Elliptic genera of level N for complete intersections

We focus on the elliptic genera of level N at the cusps of a congruence subgroup for any complete intersection. Writing the first Chern class of a complete intersection as a product of an integral coefficient c₁ and a generator of the 2nd integral cohomology group, we mainly discuss the values of the elliptic genera of level N for the complete intersection in the cases of c₁>, =, or<0, In particular, the values about the Todd genus, \widehat{A}\mathrm{-genus}, and A_k-genus can be derived from the elliptic genera of level N.     

Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold

We present simple trace formulas for Hecke operators Tk(p) for all p>3 on Sk(Γ₀(3)) and Sk(Γ₀(9)), the spaces of cusp forms of weight k and levels 3 and 9. These formulas can be expressed in terms of special values of Gaussian hypergeometric series and lend themselves to recursive expressions in terms of traces of Hecke operators on spaces of lower weight. Along the way, we show how to express the traces of Frobenius of a family of elliptic curves equipped with a 3-torsion point as special values of a Gaussian hypergeometric series over Fq, when . As an application, we use these formulas to provide a simple expression for the Fourier coefficients of η⁸(3z), the unique normalized cusp form of weight 4 and level 9, and then show that the number of points on a certain threefold is expressible in terms of these coefficients.     

Characterization of modules of finite projective dimension over complete intersections

Let be a finitely generated module over a local complete intersection of characteristic . The property that has finite projective dimension can be characterized by the vanishing of for some and for some .