Hiren Maharaj

Code construction on fiber products of Kummer covers

We show that Riemann-Roch spaces of divisors from fiber products of Kummer covers of the projective line, which are invariant with respect to the Galois group, decompose as a direct sum of Riemann-Roch spaces of divisors of the projective line. Consequently, one obtains explicit bases and good upper bounds for the minimum distance of the resulting Goppa codes. This correspondence is a generalization of the work of Xing.

New Optimal Tame Towers of Function Fields over Small Finite Fields

Ihara [11] introduced the quantity A(q) = limsupg→∞ N q (g)/g where N q (g) is the maximum number of rational places of a function field with genus g and with the finite field F q as the full field of constants. Drinfeld and Vladut [2] showed that A(q) ≤ √q - 1. It was also shown by Ihara [11], and Tsfasman, Vladut and Zink [17] in special cases, that A(q)= √ q - 1 when q is a square. When q is not a square, the exact value of A(q) is currently unknown. While the problem of finding A(q) in this case is an interesting problem in its own right, much motivation comes from implications in asymptotic results in coding theory. Essentially there are three approaches to finding lower bounds for A(q): class field towers [15], modular curves [11], [17], [3], [4]and explicit towers (that is, given explicitly in terms of generators and relations) of function fields. For applications to coding theory though, explicit towers are needed. In [6], a tower of function fields over Fq is defined to be a sequence F = (F 1, F 2, F 3,.. ) of function fields F i , having the following properties: (i) F1 ⊆F2 ⊆ F3 ⊆.... (ii) For each n ≥ 1, the extension F n+1/ F n is separable of degree [F n+1: F n ] > 1. (iii) the genus g(F j ) > 1 for some j ≥ 1. (iv) F q is the full field of constants of each F n .

Finite field elements of high order arising from modular curves

In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate that the resulting elements have high order. Between the two constructions, we are able to generate high order elements in every characteristic. Despite the use of the modular recursions of Elkies, our methods are quite elementary and require no knowledge of modular curves. We compare our results to a recent result of Voloch. In order to do this, we state and prove a slightly more refined version of a special case of his result.

On the floor and the ceiling of a divisor

Given a divisor A of a function field, there is a unique divisor of minimum degree that defines the same vector space of rational functions as A and there is a unique divisor of maximum degree that defines the same vector space of rational differentials as A. These divisors are called the floor and the ceiling of A. A method is given for finding both the floor and the ceiling of a divisor. The floor and the ceiling of a divisor give new bounds for the minimum distance of algebraic geometry codes. The floor and the ceiling of a divisor supported by collinear places of the Hermitian function field are determined. Finally, we find the exact code parameters for a large class of algebraic geometry codes constructed from the Hermitian function field.

Lattices from elliptic curves over finite fields

In their well known book 6 Tsfasman and Vladut introduced a construction of a family of function field lattices from algebraic curves over finite fields, which have asymptotically good packing density in high dimensions. In this paper we study geometric properties of lattices from this construction applied to elliptic curves. In particular, we determine the generating sets, conditions for well-roundedness and a formula for the number of minimal vectors. We also prove a bound on the covering radii of these lattices, which improves on the standard inequalities.

On the construction of tame towers over finite fields

Abstract Recently, W.-C. W. Li, et al. (Lect. Notes in Comput. Sci. 2369 (2002) 372) developed a non-deterministic algorithm to perform a computer search for polynomials that recursively define asymptotically good sequences of function fields. In this paper, we build on this work by refining this algorithm. We give many sufficient conditions for the construction of such sequences and we describe the techniques used in the search. Many examples are given. The resulting towers are important for the construction of asymptotically good sequences of codes and they could provide further numerical evidence for Elkies’ modularity conjecture.

On lattices generated by finite Abelian groups

This paper is devoted to the study of lattices generated by finite Abelian groups. Special species of such lattices arise in the exploration of elliptic curves over finite fields. In case the generating group is cyclic, they are also known as the Barnes lattices. It is shown that for every finite Abelian group with the exception of the cyclic group of order four these lattices have a basis of minimal vectors. Another result provides an improvement of a recent upper bound by Min Sha for the covering radius in the case of the Barnes lattices. Also discussed are properties of the automorphism groups of these lattices.

Explicit constructions of algebraic-geometric codes

We propose a simple construction of algebraic-geometric codes which are subcodes of Goppa codes and which coincide with Goppa codes in many cases. The codes we construct have the advantage that for an explicitly given extension of the rational function field, one can easily obtain explicit bases and therefore an exact formula for the dimension. Furthermore, we show that in many cases good upper and lower bounds for the minimum distance can be obtained

Generalized AG codes and generalized duality

In (J. Pure Appl. Algebra 55 (1988) 199), Stichtenoth gave sufficient conditions for algebraic-geometric Goppa codes to be self-dual. We adapt this work using differentials to the setting of generalized AG codes introduced by Xing et al. (IEEE Trans. Inform. Theory 45 (1999) 2498). As a consequence we are naturally led to generalized notions of duality for codes. Many examples are presented.

A Note on Further Improvements of the TVZ-Bound

In this note, we present improvements of recent asymptotic results on constructions of nonlinear codes using algebraic geometry. The main constructions involve an adaptation of recent constructions by Niederreiter and Oumlzbudak; and Stichtenoth and Xing