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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 the form x m + ax ..."
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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.
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.
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.
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.
SURVEY ARTICLE SIMPLE GROUPS AND SIMPLE LIE ALGEBRAS
, 1963
"... One of the main aims of workers in the theory of groups has always been the determination of all finite simple groups. For simple groups may be regarded as the fundamental building blocks out of which finite groups are constructed. The cyclic groups of prime order are trivial examples of simple grou ..."
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One of the main aims of workers in the theory of groups has always been the determination of all finite simple groups. For simple groups may be regarded as the fundamental building blocks out of which finite groups are constructed. The cyclic groups of prime order are trivial examples of simple groups, and are the only simple groups which are Abelian. The first examples of nonAbelian simple groups were discovered by Galois, who showed that the alternating group An is simple if n ^ 5. The group A5 of order 60 is the smallest nonAbelian simple group. Further examples of finite simple groups are the socalled classical groups, i.e. the linear, symplectic, orthogonal and unitary groups over finite fields, which were first introduced by Jordan [8] and studied in detail by Dickson [4]. These groups are defined as follows. GF(q) denotes the Galois field with q elements, where q is any prime power. (i) The linear groups. Let GLn(q) be the group of all nonsingular nxn matrices over GF(q), SLn(q) the subgroup of matrices of determinant 1, and PSLn(q) the factor group of SLn(q) by its centre. Then PSLn(q) is simple when n^2. (ii) The symplectic groups. Let V be a finite dimensional vector space over GF(q) and f(x, y) be a nonsingular bilinear form on V with values in GF{q) satisfying f(x, x) = 0 for all xeV. Then / is skewsymmetiic, i.e. f(y, x) — —f{x, y) for all x,yeV. The existence of such a bilinear form implies that the dimension of V is even. Let dim V = 2n. Then a basis 2n 2n el5 e2,..., e2n can be chosen for F such that, if x = 2 & ^ y — S ^ ei} then •i=i i=i n /(« » V) =.2 (£< Vn+i Vi £n+i)Let Sp2n(q) be the group of all nonsingular linear transformations of V into itself satisfying f(x, y)=f(Tx, Ty), and let PSp2n(q) be the factor group of Sp2n{q) by its centre. Then PSp2n(q) is simple and PSp2(q) is isomorphic to PSL2(q). Received 9 December, 1964. The present survey article is an expanded version of a lecture delivered at the meeting of the British Mathematical Colloquium in
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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