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Permutation binomials over finite fields
 TRANS. AMER. MATH. SOC
, 2007
"... We prove that, if x m + ax n permutes the prime field Fp, where m> n> 0 and a ∈ F ∗ p, then gcd(m − n, p − 1)> √ p − 1. Conversely, we prove that if q ≥ 4 and m> n> 0 are fixed and satisfy gcd(m − n, q − 1)> 2q(log log q) / log q, then there exist permutation binomials over Fq of ..."
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We prove that, if x m + ax n permutes the prime field Fp, where m> n> 0 and a ∈ F ∗ p, then gcd(m − n, p − 1)> √ p − 1. Conversely, we prove that if q ≥ 4 and m> n> 0 are fixed and satisfy gcd(m − n, q − 1)> 2q(log log q) / log q, then there exist permutation binomials over Fq of the form x m + ax n if and only if gcd(m, n, q − 1) = 1.
Some families of permutation polynomials over finite fields, Int
 Hill Center, Department of Mathematics, Rutgers University
"... Abstract. We give necessary and sufficient conditions for a polynomial of the form x r (1 + x v + x 2v + · · · + x kv) t to permute the elements of the finite field Fq. Our results yield especially simple criteria in case (q − 1) / gcd(q − 1, v) is a small prime. 1. ..."
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Abstract. We give necessary and sufficient conditions for a polynomial of the form x r (1 + x v + x 2v + · · · + x kv) t to permute the elements of the finite field Fq. Our results yield especially simple criteria in case (q − 1) / gcd(q − 1, v) is a small prime. 1.
CLASSES OF PERMUTATION POLYNOMIALS BASED ON CYCLOTOMY AND AN ADDITIVE ANALOGUE
"... Abstract. I present a construction of permutation polynomials based on cyclotomy, an additive analogue of this construction, and a generalization of this additive analogue which appears to have no multiplicative analogue. These constructions generalize recent results of José Marcos. Dedicated to Mel ..."
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Abstract. I present a construction of permutation polynomials based on cyclotomy, an additive analogue of this construction, and a generalization of this additive analogue which appears to have no multiplicative analogue. These constructions generalize recent results of José Marcos. Dedicated to Mel Nathanson on the occasion of his sixtieth birthday 1.
ON SOME PERMUTATION POLYNOMIALS OVER Fq OF THE FORM x r h(x (q−1)/d)
"... Abstract. In a recent paper, Akbary and Wang gave a sufficient condition for x u + x r to permute Fq, in terms of the period of a certain sequence involving sums of cosines. As an application they gave necessary and sufficient conditions in case u, r, q satisfy certain special properties. We show th ..."
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Abstract. In a recent paper, Akbary and Wang gave a sufficient condition for x u + x r to permute Fq, in terms of the period of a certain sequence involving sums of cosines. As an application they gave necessary and sufficient conditions in case u, r, q satisfy certain special properties. We show that the AkbaryWang sufficient condition follows from a more general sufficient condition which does not involve sums of cosines. This leads to vastly simpler proofs of the AkbaryWang results, as well as generalizations to polynomials of the form x r h(x (q−1)/d). 1.
Coding theory and algebraic combinatorics
, 2008
"... This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In part ..."
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This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.
United States of America, "Plaintiff v
 USA
, 1992
"... One of the best ways to understand the nature of finite simple groups is through geometries associated with them. This approach for classical and exceptional groups of Lie type has been quite successful and has led to the deveopment of the concept of buildings and polar spaces. The latter have been ..."
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One of the best ways to understand the nature of finite simple groups is through geometries associated with them. This approach for classical and exceptional groups of Lie type has been quite successful and has led to the deveopment of the concept of buildings and polar spaces. The latter have been characterized by simple systems of axioms with a combinatorialgeometric flavour. It has been observed recently that geometries similar to buildings can be associated with finite sporadic simple groups. However, most of the known characterizations of such geometries for sporadic groups require additional assumptions of a grouptheoretic nature. One aim of this thesis is to present characterizations of geometries for the sporadic groups J2, Suz, McL, Co3, Fi22, Fi23, Fi24 and He, which are in the same spirit as the characterizations of buildings and polar spaces mentioned above, in particular without any assumption on the automorphism groups of the geometries. A byproduct of these results for J2, Suz and He is a proof that certain presentations for those groups are faithful. Most of this work may be viewed as a contribution to the theory of graphs
The Classification of the Finite Simple Groups: An Overview
 MONOGRAFÍAS DE LA REAL ACADEMIA DE CIENCIAS DE ZARAGOZA. 26: 89–104, (2004)
, 2004
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