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An Invitation to Computational Group Theory
 Groups' 93  Galway/St. Andrews, volume 212 of London Math. Soc. Lecture Note Ser
, 1995
"... Algebra" in 1967 [Lee70]. Its proceedings contain a survey of what had been tried until then [Neu70] but also some papers that lead into the Decade of discoveries (19671977). At the Oxford conference some of those computational methods were presented for the first time that are now, in some cases ..."
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Algebra" in 1967 [Lee70]. Its proceedings contain a survey of what had been tried until then [Neu70] but also some papers that lead into the Decade of discoveries (19671977). At the Oxford conference some of those computational methods were presented for the first time that are now, in some cases varied and improved, work horses of CGT systems: Sims' methods for handling big permutation groups [Sim70], the KnuthBendix method for attempting to construct a rewrite system from a presentation [KB70], variations of the ToddCoxeter method for the determination of presentations of subgroups [Men70]. Others, like J. D. Dixon's method for the determination of the character table [Dix67], the pNilpotentQuotient method of I. D. Macdonald [Mac74] and the ReidemeisterSchreier method of G. Havas [Hav74] for subgroup presentations were published within a few years from that conference. However at least equally important for making group theorists aware of CGT were a number of applications of...
Regular Hyperbolic Fibrations
 MR 2002f:51018 Zbl 0991.51006
"... A hyperbolic bration is a set of q 1 hyperbolic quadrics and two lines which together partition the points of PG(3; q). The classical example of a hyperbolic bration comes from a pencil of quadrics; however, several other families are now known. In this paper we begin the development of a gene ..."
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A hyperbolic bration is a set of q 1 hyperbolic quadrics and two lines which together partition the points of PG(3; q). The classical example of a hyperbolic bration comes from a pencil of quadrics; however, several other families are now known. In this paper we begin the development of a general framework to study hyperbolic brations for odd prime powers q.
Simultaneous Constructions of the Sporadic Groups Co2
, 906
"... Abstract. In this article we give selfcontained existence proofs for the sporadic simple groups Co2 and Fi22 using the second author’s algorithm [10] constructing finite simple groups from irreducible subgroups of GLn(2). These two sporadic groups were originally discovered by J. Conway [4] and B. ..."
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Abstract. In this article we give selfcontained existence proofs for the sporadic simple groups Co2 and Fi22 using the second author’s algorithm [10] constructing finite simple groups from irreducible subgroups of GLn(2). These two sporadic groups were originally discovered by J. Conway [4] and B. Fischer [7], respectively, by means of completely different and unrelated methods. In this article n = 10 and the irreducible subgroups are the Mathieu group M22 and its automorphism group Aut(M22). We construct their five nonisomorphic extensions Ei by the two 10dimensional nonisomorphic simple modules of M22 and by the two 10dimensional simple modules of A22 = Aut(M22) over F = GF(2). In two cases we construct the centralizer Hi = CG i (zi) of a 2central involution zi of Ei in any target simple group Gi. Then we prove that all the conditions of Algorithm 7.4.8 of [11] are satisfied. This allows us to construct G3 ∼ = Co2 inside GL23(13) and G2 ∼ = Fi22 inside GL78(13). We also calculate their character tables and presentations. 1.
Projections of Binary Linear Codes onto Larger Fields
, 2003
"... We study certain projections of binary linear codes onto larger fields. These projections include the wellknown projection of the extended Golay [24, 12, 8] code onto the Hexacode over GF(4) and the projection of the ReedMuller code R(2, 5) onto the unique selfdual [8, 4, 4] code over GF(4). We g ..."
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We study certain projections of binary linear codes onto larger fields. These projections include the wellknown projection of the extended Golay [24, 12, 8] code onto the Hexacode over GF(4) and the projection of the ReedMuller code R(2, 5) onto the unique selfdual [8, 4, 4] code over GF(4). We give a characterization of these projections, and we construct several binary linear codes which have best known optimal
Gauss sums, Jacobi sums, and pranks of cyclic difference sets
 Journal of Combinatorial Theory Series A
, 1999
"... We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2 d ..."
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We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2 d
Shult sets and translation ovoids of the Hermitian surface
 Combinatoria 15E (Gennaio 2005), 1–17, Università degli Studi di Roma “La Sapienza”, Dipartimento di Matematica, Advances in Geometry
"... Starting with carefully chosen sets of points in the Desarguesian affine plane AG(2, q 2) and using an idea first formulated by E. Shult, several infinite families of translation ovoids of the Hermitian surface are constructed. Various connections with locally Hermitian 1–spreads of Q − (5, q) and s ..."
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Starting with carefully chosen sets of points in the Desarguesian affine plane AG(2, q 2) and using an idea first formulated by E. Shult, several infinite families of translation ovoids of the Hermitian surface are constructed. Various connections with locally Hermitian 1–spreads of Q − (5, q) and semifield spreads of P G(3, q) are also discussed. Finally, geometric characterization results are developed for the translation ovoids arising in the so–called classical and semiclassical settings.
The GAP 4 Type System Organising Algebraic Algorithms
"... Version 4 of the GAP (Groups, Algorithms, Programming) system for computational discrete mathematics has a number of novel features. In this paper, we describe the type system, and the way in which it is used for method selection. This system is central to the organisation of the library which is th ..."
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Version 4 of the GAP (Groups, Algorithms, Programming) system for computational discrete mathematics has a number of novel features. In this paper, we describe the type system, and the way in which it is used for method selection. This system is central to the organisation of the library which is the main part of the GAP system. Unlike simpler objectoriented systems, GAP allows method selection based on the types of all arguments and on certain aspects of the relationship between the arguments. In addition, the type of an object can change, in a controlled way, during its life. This reflects information about the object which has been computed and stored. Individual methods can be written and installed independently. Furthermore most checking of the arguments is done in a uniform way by the method selection system, making individual methods simpler and less prone to error. The methods are combined automatically to produce a powerful and usable system for interactive use or programming...
Polynomial Codes and Finite Geometries
, 1996
"... Contents 1 Introduction 2 Projective and affine geometries 3 2.1 Projective geometry ....................... 3 2.2 Arline geometry .......................... 7 2.3 Designs from geometries ..................... 10 2.4 Codes from designs ........................ 11 The 3.1 3.2 3.3 ReedMuller cod ..."
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Contents 1 Introduction 2 Projective and affine geometries 3 2.1 Projective geometry ....................... 3 2.2 Arline geometry .......................... 7 2.3 Designs from geometries ..................... 10 2.4 Codes from designs ........................ 11 The 3.1 3.2 3.3 ReedMuller codes 12 Geometries and ReedMuller codes ............... 16 Decoding ............................. 22 4 The groupalgebra approach 25 4.1 Elementary results and Berman's theorem ........... 26 4.2 Isometries of the group algebra ................. 28 4.3 Translationinvariant extended cyclic codes .......... 30 4.4 The generator polynomials of punctured ReedMuller codes and their pary analogues .................... 33 4.5 Orthogonals and annihilators .................. 36 *The authors wish to thank Paul Camion, Pascale Charpin and Projet Codes at INRIA for the hospitality and support shown during the preparation of this manuscript. In particular, the first author spent much of 19921993 at
SKEW HADAMARD DIFFERENCE SETS FROM THE REETITS SLICE SYMPLECTIC SPREADS IN
, 2006
"... Abstract. Using a class of permutation polynomials of F 3 2h+1 obtained from the ReeTits slice symplectic spreads in PG(3,3 2h+1), we construct a family of skew Hadamard difference sets in the additive group of F 3 2h+1. With the help of a computer, we show that these skew Hadamard difference sets ..."
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Abstract. Using a class of permutation polynomials of F 3 2h+1 obtained from the ReeTits slice symplectic spreads in PG(3,3 2h+1), we construct a family of skew Hadamard difference sets in the additive group of F 3 2h+1. With the help of a computer, we show that these skew Hadamard difference sets are new when h = 2 and h = 3. We conjecture that they are always new when h> 3. Furthermore, we present a variation of the classical construction of the twin prime power difference sets, and show that inequivalent skew Hadamard difference sets lead to inequivalent difference sets with twin prime power parameters. 1.
Caps Embedded in Grassmannians
 Geom. Dedicata
, 1998
"... This paper is concerned with constructing caps embedded in line Grassmannians. In particular, we construct a cap of size q 3 + 2q 2 + 1 embedded in the Klein quadric of PG(5; q) for even q, and show that any cap maximally embedded in the Klein quadric which is larger than this one must have size ..."
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This paper is concerned with constructing caps embedded in line Grassmannians. In particular, we construct a cap of size q 3 + 2q 2 + 1 embedded in the Klein quadric of PG(5; q) for even q, and show that any cap maximally embedded in the Klein quadric which is larger than this one must have size equal to the theoretical upper bound, namely q 3 + 2q 2 + q + 2. It is not known if caps achieving this upper bound exist for even q ? 2. 1 Introduction In [7] Glynn showed that any full Singer line orbit in PG(3; q) corresponds to a cap of size q 3 +q 2 +q+1 embedded in the Klein quadric K of PG(5; q). Moreover, for odd q he observed that this is the largest possible cap embedded in K. In this paper we show that larger caps can be embedded in K for even q, and we explicitly construct several infinite families of caps maximally embedded in K. The problem of completing caps to maximum caps on K is also addressed. Finally, we extend Glynn's idea to higher dimensions, thereby constru...