Aiden A. Bruen

Intersections in Projective Space II

A complete classification is given of pencils of quadrics in projective space of three dimensions over a finite field, where each pencil contains at least one non-singular quadric and where the base curve is not absolutely irreducible. This leads to interesting configurations in the space such as partitions by elliptic quadrics and by lines.

The n-queens problem

We present some new solutions to the problem of arranging n queens on an n x n chessboard with no two taking each other. Recent related work of other authors is also discussed.

Arcs and blocking sets II

In a finite projective plane 1t a blocking set is a set S of points such that each line contains at least one point in S and at least one point not in S. The main results in this note are Theorems 1.1 and 1.7 and Corollaries 1.8 and 1.9. Theorem 1.1 describes new bounds on certain kinds of reduced blocking sets in PG(2, q). Theorem 1.7 and Corollary 1.8 give new bounds on the cardinality of a reduced (and so, of an arbitrary) blocking set Sin PG(2, q). These bounds yield a significant improvement on previously known results. The proof of 1.7 uses a combinatorial argument together with special cases of some deep results in Jamison [9] and R6dei [10]. A fortuitous factorization makes the result more tractable. (The result of 1.1 is used in 1.4 to obtain bounds on the size of complete arcs in PG(2, q): in some cases these results give a slight improvement on the results in Hirschfeld [8]). Corollary 1.9 shows how the structure ofPG(2, q) is being utilized: it yields a far stronger result than a related result for general planes in Bruen and Thas [5] (the case n > 4 in Theorem 3 there). For blocking sets in arbitrary finite projective planes not much is known apart from 2.2. Here we offer a new proof based on an idea in Hill and Mason [7]. Moreover the proof can be generalized to arbitrary 2-designs as in Theorem 2.3. Fundamental to the improved bound on lSI in PG(2, q) is a result of Jamison [9] on intersection sets in the classical affine plane AG(2, q). His result is not valid for general finite affine planes: the problem of finding good bounds on the size of blocking sets in finite affine planes is open. Our result (Theorem 3.1) describes the best known such bounds. We use the fact that an affine plane (which is of course a 2-design) also has its lines arranged into parallel classes, so our result is a slight improvement on Theorem 2.3 in the case of finite affine planes.

Binary codes and caps

The connection between maximal caps (sometimes called complete caps) and certain binary codes called quasi-perfect codes is described. We provide a geometric approach to the foundational work of Davydov and Tombak who have obtained the exact possible sizes of large maximal caps. A new self-contained proof of the existence and the structure of the largest maximal nonaffine cap in ℙG(n, 2) is given. Combinatorial and geometric consequences are briefly sketched. Some of these, such as the connection with families of symmetric-difference free subsets of a finite set will be developed elsewhere.

Arcs in PG(n, q), MDS-codes and three fundamental problems of B. Segre — Some extensions

To each arc of PG(n, q) an algebraic hypersurface is associated. Using this tool new results on complete arcs are obtained. Since arcs and linear MDS-codes are equivalent objects, these results can be translated in terms of codes.

Maximum distance separable codes and arcs in projective spaces

Given any linear code C over a finite field GF(q) we show how C can be described in a transparent and geometrical way by using the associated Bruen-Silverman code. Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes. Our proofs make use of the connection between the work of Redei [L. Redei, Lacunary Polynomials over Finite Fields, North-Holland, Amsterdam, 1973. Translated from the German by I. Foldes. [18]] and the Redei blocking sets that was first pointed out over thirty years ago in [A.A. Bruen, B. Levinger, A theorem on permutations of a finite field, Canad. J. Math. 25 (1973) 1060-1065]. The main results of this paper significantly strengthen those in [A. Blokhuis, A.A. Bruen, J.A. Thas, Arcs in PG(n,q), MDS-codes and three fundamental problems of B. Segre-Some extensions, Geom. Dedicata 35 (1-3) (1990) 1-11; A.A. Bruen, J.A. Thas, A.Blokhuis, On M.D.S. codes, arcs in PG(n,q) with q even, and a solution of three fundamental problems of B. Segre, Invent. Math. 92 (3) (1988) 441-459].

The minimal number of lines intersected by a set of q +2 points, blocking sets, and intersecting circles

Lower bounds are given for the number of lines blocked by a set of q + 2 points in a projective plane of order q. Implications are discussed to the theory of blocking sets and bounds are obtained for the size of a double intersecting set of circles in a Mobius plane.

Long Binary Linear Codes and Large Caps in Projective Space

We obtain, in principle, a complete classification of all long inextendable binary linear codes. Several related constructions and results are presented.

Intersections in projective space I: Combinatorics

On obtient par des arguments combinatoires de denombrement la classification des intersections dans un espace projectif a n dimensions sur un corps

Polynomials with Frobenius groups of prime degree as Galois groups II

Abstract We present characterization theorems for polynomials of prime degree p ≥ 5 over Q with Frobenius groups of degree p: Fpl = Flp′, l| p − 1 as Galois groups. We construct a generic family of polynomials with Galois group F p(p − 1) 2 (p ≡ 3 (mod 4)), using the pth Chebyshev polynomial of the first kind. In addition, a parametric family of quintic polynomials with Galois group F20 is given. Explicit examples of polynomials are presented for F2p = Dp (p ≤ 19), F20 and for F p(p − 1) 2 with p = 7 and 11. Some properties of Frobenius fields are also discussed.