Abstract

We discuss a number of open problems and conjectures in the theory and application of finite fields. We also provide a brief discussion of the status as well as references related to each problem.

Introduction

In this paper we try to summarise some interesting and/or important questions in the theory and application of finite fields. These questions obviously reflect our personal tastes but we have indeed tried to consider questions of general interest. We hope that these questions and even more, the methods developed for their solutions, will be of interest to other researchers. The reader may wonder why some questions we call ‘Problems’ and some we call ‘Conjectures’. Roughly speaking, we use the term Conjecture if we (and very often others) believe the statement to be true while the term Problem is used to indicate all other statements. In general we feel that our conjectures may be more difficult to resolve than our problems.

The standard notions of the theory of finite fields which we use can be found in the finite field bible by Lidl and Niederreiter.

Combinatorics

There are numerous open problems in combinatorics which are related to finite fields. In this section we briefly describe several of these. We begin with several questions related to latin squares. A latin square of order n is an n × n array based upon n distinct symbols with the property that each row and each column contains each of the n symbols exactly once.

This volume represents the refereed proceedings of the Third International Conference on Finite Fields and Applications held at the University of Glasgow, Scotland, 11–14 July, 1995, where it was hosted by the Department of Mathematics. The conference, often referred to as Fq3, was the successor of two other international conferences concerning finite fields held at the University of Nevada, Las Vegas, USA, in August 1991 and in August 1993. The Organising Committee comprised Steve Cohen, Stuart Hoggar, Bob Odoni (all of the University of Glasgow), James Hirschfeld (University of Sussex), Gary Mullen (Pennsylvania State University), Harald Niederreiter (Austrian Academy of Sciences) and Peter Shiue (University of Nevada, Las Vegas).

Finite fields with their tight structure are intrinsically fascinating; further, their study is now recognised to be extremely useful in diverse areas of pure and applicable mathematics, including aspects of number theory, algebra, analysis and algebraic geometry and, at the same time, manifold aspects of information theory, computer science and engineering. For example, coding theory is enriched by deep ideas, crucially involving finite fields, on exponential sums, function fields and linear algebra, and in turn, has stimulated further questions for finite field research. Indeed, what is particularly exciting in current activity is the interplay between various areas, within pure mathematics itself and within those having definite applications. A further sign of this vitality is the emergence in 1995 of the journal Finite Fields and Their Applications published by Academic Press.

Abstract. We give a survey on a topic in Finite Geometry which has generated considerable interest in the literature: the construction of maximal sets of mutually orthogonal Latin squares (MOLS) or, equivalently, of maximal nets. Most known constructions depend on finite fields either directly or via Galois geometry. Our subject splits naturally in two parts, namely the existence problem for small and large maximal sets of MOLS, respectively; in the first case, difference matrix methods have proved to be particularly useful, whereas the second case rests almost completely on the study of maximal partial t-spreads in finite protective spaces. For this reason, we also give a short review of what is known on the existence of maximal partial t-spreads.

INTRODUCTION

In what follows, we shall give a survey on a topic in Finite Geometry which has generated considerable interest in the literature: the construction of maximal sets of mutually orthogonal Latin squares (MOLS) or, equivalently, of maximal nets. As is well-known, the existence of a (Bruck) net of order s and degree r (for short, an (s, r)-net)) is equivalent to that of a set of r–2 mutually orthogonal Latin squares of order s; it is also well-known that this correspondence respects maximality as well as the stronger property of being transversal-free.

Abstract

New C++-implementations of the classical factorization algorithms for polynomials over finite fields of Berlekamp and the new ones of Niederreiter are presented. Their performances on various types of inputs are compared.

Introduction

The basic problem of factorizing univariate polynomials over the finite field Fq has got new impulses in the past few years with a new linearization technique developed by Niederreiter in [8], [9], [10]. Unlike Berlekamp's classical approach, which uses the Frobenius fixed point algebra in A := Fq[X]/(f) (where f is the polynomial to be factored), Niederreiter's method is based on the analysis of the solution space of certain differential equations in the field of rational functions Fq(X).

From the very beginning there have been several striking similarities between Niederreiter's and Berlekamp's algorithms in each step. Suppose for simplicity that the polynomial is monic and squarefree. Then in both algorithms a certain system of linear equations has to be set up and solved, leading to an Fq - subspace S of A, whose dimension coincides with the number of irreducible factors of f. Now the elements of S can be used to extract the irreducible factors of f by suitable gcd operations.

Niederreiter's algorithm has the following practical advantages: In the case of small fields the linear equations to be solved can be set up very efficiently. In particular in F2 they can be read off directly from the coefficients of f.

Abstract. This paper is a working out of the same-titled talk given by the author at the Third International Conference on Finite Fields and Their Applications in Glasgow, 1995. We give a survey on recent results on the characterization, the structure, the enumeration, and the construction of completely free elements and normal bases in finite dimensional extensions over finite fields.

A Strengthening of the Normal Basis Theorem. If E is a finite dimensional Galois extension over a field F with Galois group G, then the Normal Basis Theorem states that the additive group (E, +) of E is a cyclic module over the group algebra FG, i.e., there exists an element w in E such that the set {g(w) | g ∈ G} of G-conjugates of w is an F-basis of E. Such a basis is called a normal basis in E over F. Every generator w of E as FG-module is called a normal basis generator in E over F. For the sake of simplicity such an element is also called free in E over F.

If H is a subgroup of G, and Fix(H) is the intermediate field of E over F belonging to H via the Galois correspondence, i.e., the subfield of E which is fixed elementwise by H, then (E, +) likewise carries the structure of a Fix(H) H-module.

Abstract – A general algebraic method for decoding all cyclic codes up to their actual minimum distance d is presented. Full error-correcting capabilities t = [(d − 1)/2] of the codes are therefore achieved. In contrast to the decoding method recently suggested by Chen et. al., our method uses for the first time characteristic sets instead of Gröbner bases as the algebraic tool to solve the system of multivariate syndrome equations. The characteristic sets method is generally faster than the Gröbner bases method.

A new strategy called “Fill-Holes” method is also presented. It uses Gröbner bases or characteristic sets to find certain unknown syndromes and then combines the computational methods with the well-implemented BCH decoding algorithm.

Keywords – Coding theory, cyclic codes, decoding, ideal theory, Gröbner bases, characteristic sets.

INTRODUCTION

One important objective in coding theory has always been the construction of algebraic algorithms, that are capable of decoding all cyclic codes up to their actual minimum distance. Full error-correcting capabilities of the codes can only be achieved when such algorithms are available. For many years, algebraic decoding of cyclic codes has been constrained by the lower bound on the minimum distance of the codes. For example, the commonly used Berlekamp-Massey algorithm is known to be restricted within the BCH bound when it is used to decode all cyclic codes. Such restrictions can be traced to the fact that the algorithm requires syndromes to be contiguous in the Newton's identities.

ABSTRACT. We present a survey of recent work of the authors in which sequences of quasirandom points are constructed by new methods based on global function fields. These methods yield significant improvements on all earlier constructions. The most powerful of these methods employ global function fields with many rational places, or equivalently algebraic curves over finite fields with many rational points. With the help of class field theory for global function fields, it can be shown that our constructions are best possible in the sense of the order of magnitude of quality parameters. The paper contains also a new construction of sequences of quasirandom points and new facts about the earlier constructions designed by the authors.

Introduction

The motivation for the work that we want to present here stems from the theory of uniform distribution of sequences in number theory and from quasi-Monte Carlo methods in numerical analysis. A key problem in these areas is how to distribute points as uniformly as possible over an s-dimensional unit cube Is = [0, 1]s, s ≥ 1. A precise formulation of this problem will be given below. The essence of our work is that methods based on global function fields (or, equivalently, on algebraic curves over finite fields) yield excellent constructions of (finite) point sets and (infinite) sequences with strong uniformity properties. In fact, these methods are so powerful that they lead to constructions which are, in a sense to be explained later, best possible.