Gary L. Mullen

Discrete mathematics using Latin squares

LATIN SQUARES. A Brief Introduction to Latin Squares. Mutually Orthogonal Latin Squares. GENERALIZATIONS. Orthogonal Hypercubes. Frequency Squares. RELATED MATHEMATICS. Principle of Inclusion--Exclusion. Groups and Latin Squares. Graphs and Latin Squares. APPLICATIONS. Affine and Projective Planes. Orthogonal Hypercubes and Affine Designs. Magic Squares. Room Squares. Statistics. Error--Correcting Codes. Cryptology. (t,m,s)--Nets. Miscellaneous Applications of Latin Squares. Appendices. Indexes.

Handbook of Finite Fields

Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and each chapter is self contained and peer reviewed. The first part of the book traces the history of finite fields through the eighteenth and nineteenth centuries. The second part presents theoretical properties of finite fields, covering polynomials, special functions, sequences, algorithms, curves, and related computational aspects. The final part describes various mathematical and practical applications of finite fields in combinatorics, algebraic coding theory, cryptographic systems, biology, quantum information theory, engineering, and other areas. The book provides a comprehensive index and easy access to over 3,000 references, enabling you to quickly locate up-to-date facts and results regarding finite fields.

An Equivalence between (T, M, S)-Nets and Strongly Orthogonal Hypercubes

It is well known that (t, m, s)-nets are useful in numerical analysis. While many of the best constructions of such nets arise from number theoretic or algebraic constructions, we will show in this paper that the existence of a (t, t+k, s)-net in basebis equivalent to the existence of a set of s strongly orthogonal hypercubes of dimensiont+k, orderband strengthk. Fork>2 such generalized orthogonal hypercubes provide new combinatorial structures that may be of interest in various other combinatorial settings.

Parallel computing of a quasi-Monte Carlo algorithm for valuing derivatives

Abstract The performance of the standard Monte Carlo method is compared with the performance obtained through the use of ( t , m , s )-nets in base b in the approximation of several high dimensional integral problems in valuing derivatives and other securities. The ( t , m , s )-nets are generated by a parallel algorithm, where particular considerations are given to scalability of dynamic adaptive routing and load balancing in the design and implementation of the algorithm. From the numerical evidence it appears that such nets can be powerful tools for valuing such securities.

Updated tables of parameters of (T, M, S)-nets

We present an updated survey of the known constructions and bounds for (t;m;s)- nets as well as tables of upper and lower bounds on their parameters for various bases. c 1999

Polynomial representation of complete sets of mutually orthogonal frequency squares of prime power order

Construction des ensembles de carres de frequences mutuellement orthogonaux, basee sur les polynomes definis sur les corps finis

Distribution of irreducible polynomials of small degrees over finite fields

D. Wan very recently proved an asymptotic version of a conjecture of Hansen and Mullen concerning the distribution of irreducible polynomials over finite fields. In this note we prove that the conjecture is true in general by using machine calculation to verify the open cases remaining after Wans work.

Self-Reciprocal Irreducible Polynomials Over Finite Fields

We classify monic self-reciprocal irreducible polynomials over finite fields in terms of their orders. We also study the weights of these polynomials.

Figures of merit for digital multistep pseudorandom numbers

The statistical independence properties of s successive digital multistep pseudorandom numbers are governed by the figure of merit p(s) (I) which depends on s and the characteristic polynomial f of the recursion used in the generation procedure. We extend previous work for s = 2 and describe how to obtain large figures of merit for s > 2, thus arriving at digital multistep pseudorandom numbers with attractive statistical independence properties. Tables of figures of merit for s = 3, 4, 5 and degrees < 32 are included.

Value sets of Dickson polynomials over finite fields

Abstract Let Fq denote the finite field of order q where q is a prime power. If a ∈ Fq and d ≥ 1 is an integer, define the Dickson polynomial g d (x, a) = ∑ t=0 [ d 2 ] ( d (d−t) )( t d−t )(−a t x d−2t . Let {gd(x, a) | x ∈ Fq} denote the image or value set of the polynomial gd(x, a). In this paper we determine the cardinality of the value set for the Dickson polynomial gd(x, a) over the finite field Fq.