We study certain projections of binary linear codes onto larger fields. These projections include the well-known projection of the extended Golay [24, 12, 8] code onto the Hexacode over GF(4) and the projection of the Reed-Muller code R(2, 5) onto the unique self-dual [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

The Catalan monoid and partial Catalan monoid of a directed graph are introduced.

We study sets of lines of AG(n, q) and P G(n, q) with the property that no three lines form a triangle. As a result the associated point-line incidence graph contains no 6-cycles and necessarily has girth at least 8. One can then use the associated incidence matrices to form binary linear codes which can be considered as LDPC codes. The relatively high girth allows for efficient implementation of these codes. We give two general constructions for such triangle-free line sets and give the parameters for the associated codes when q is small. 1

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.

We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2 d

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 object-oriented 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...

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...

this article. 0747--7171/90/000000 + 00 $03.00/0 c fl 1996 Academic Press Limited 2 E. M. Luks, F. R'ak'oczi and C. R. B. Wright restricted classes of groups. We consider here the normalizer problem and the related conjugator problem---given groups G, H and E, determine whether there is a g in G with

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 Reed-Muller codes 12 Geometries and Reed-Muller codes ............... 16 Decoding ............................. 22 4 The group-algebra approach 25 4.1 Elementary results and Berman's theorem ........... 26 4.2 Isometries of the group algebra ................. 28 4.3 Translation-invariant extended cyclic codes .......... 30 4.4 The generator polynomials of punctured Reed-Muller codes and their p-ary 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 1992-1993 at

Abstract. Using a class of permutation polynomials of F 3 2h+1 obtained from the Ree-Tits 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.