Symplectic semifield spreads of <Emphasis Type="Italic">PG</Emphasis>(5, <Emphasis Type="Italic">q</Emphasis>) and the veronese surface

In this paper we show that starting from a symplectic semifield spread S{\mathcal{S}} of PG(5, q), q odd, another symplectic semifield spread of PG(5, q) can be obtained, called the symplectic dual of S{\mathcal{S}}, and we prove that the symplectic dual of a Desarguesian spread of PG(5, q) is the symplectic semifield spread arising from a generalized twisted field. Also, we construct a new symplectic semifield spread of PG(5, q) (q = s ², s odd), we describe the associated commutative semifield and deal with the isotopy issue for this example. Finally, we determine the nuclei of the commutative pre-semifields constructed by Zha et al. (Finite Fields Appl 15(2):125–133, 2009).     

<Emphasis Type="Italic">mth</Emphasis>-Root subgeometry partitions

New subgeometry partitions of PG(n − 1, q ^m ) by subgeometries isomorphic to PG(n − 1, q) are constructed.      

Blocking sets in PG(2, q n ) from cones of PG(2n, q)

Let Ω and be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and . Denote by K the cone of vertex Ω and base and consider the point set B defined by

Kakeya-type sets in finite vector spaces

For a finite vector space V and a nonnegative integer r≤dim V, we estimate the smallest possible size of a subset of V, containing a translate of every r-dimensional subspace. In particular, we show that if K⊆V is the smallest subset with this property, n denotes the dimension of V, and q is the size of the underlying field, then for r bounded and r<n≤rq ^r−1, we have |V∖K|=Θ(nq ^n−r+1); this improves the previously known bounds |V∖K|=Ω(q ^n−r+1) and |V∖K|=O(n ² q ^n−r+1).     

Counting the number of twin Niven numbers

Given an integer q≥2, we say that a positive integer is a q-Niven number if it is divisible by the sum of its digits in base q. Given an arbitrary integer r∈[2,2q], we say that (n,n+1,…,n+r−1) is a q-Niven r -tuple if each number n+i, for i=0,1,…,r−1, is a q-Niven number. We show that there exists a positive constant c=c(q,r) such that the number of q-Niven r-tuples whose leading component is <x is asymptotic to cx/(log x) ^r as x→∞. Research of J.M. De Koninck supported in part by a grant from NSERC. Research of I. Kátai supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA.     

Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ<Superscript>3</Superscript>

Symplectic instanton vector bundles on the projective space ℙ³ constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I _n;r of rank-2r symplectic instanton vector bundles on ℙ³ with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I _n;r ^* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).     

On the code generated by the incidence matrix of points and hyperplanes in <Emphasis Type="Italic">PG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">q</Emphasis>) and its dual

In this paper, we study the p-ary linear code C(PG(n,q)), q = p ^h , p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2,q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar’s lower bound (Geom Dedicata 8:407–415, 1979). G. Van de Voorde’s research was supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).     

On the maximality of linear codes

We show that if a linear code admits an extension, then it necessarily admits a linear extension. There are many linear codes that are known to admit no linear extensions. Our result implies that these codes are in fact maximal. We are able to characterize maximal linear (n, k, d) _q -codes as complete (weighted) (n, n − d)-arcs in PG(k − 1, q). At the same time our results sharply limit the possibilities for constructing long non-linear codes. The central ideas to our approach are the Bruen-Silverman model of linear codes, and some well known results on the theory of directions determined by affine point-sets in PG(k, q).      

A generalization of a result of Dembowski and Wagner

The famous Dembowski-Wagner theorem gives various characterizations of the classical geometric 2-design PG _n-1(n, q) among all 2-designs with the same parameters. One of the characterizations requires that all lines have size q + 1. It was conjectured [2] that this is also true for the designs PG _d (n, q) with 2 ≤ d ≤  n − 1. We establish this conjecture, hereby improving various previous results.     

Unitals in Finite Desarguesian Planes

We show that a suitable 2-dimensional linear system of Hermitian curves of PG(2,q ²) defines a model for the Desarguesian plane PG(2,q). Using this model we give the following group-theoretic characterization of the classical unitals. A unital in PG(2,q ²) is classical if and only if it is fixed by a linear collineation group of order 6(q + 1)² that fixes no point or line in PG(2,q ²).     

Blocking Sets and Derivable Partial Spreads

We prove that a GF(q)-linear Rédei blocking set of size q ^t + q ^t–1 + ··· + q + 1 of PG(2,q ^t) defines a derivable partial spread of PG(2t – 1, q). Using such a relationship, we are able to prove that there are at least two inequivalent Rédei minimal blocking sets of size q ^t + q ^t–1 + ··· + q + 1 in PG(2,q ^t), if t 4.     

Asymptotic formulas for the enumerator of trees with a given number of hanging or internal vertices

Let t(r, n) be the number of trees with n vertices of which r are hanging and q are internal (r=n−9). For a fixed r or q we prove the validity of the asymptotic formulas (r > 2)t(r, n)≈t/r|(r−2)| 2^2−r n ^2r−4 (n→∞)t(n−q, n)≈1/q|(q&#x2212;1)|q ^q−2 n ^q−1 (n→∞) In the derivation of these formulas we do not use the expression for the enumerator of the trees with respect to the number of hanging vertices. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 65–70, January, 1977.     

Minimal full polarized embeddings of dual polar spaces

Let Δ be a thick dual polar space of rank n ≥ 2 admitting a full polarized embedding e in a finite-dimensional projective space Σ, i.e., for every point x of Δ, e maps the set of points of Δ at non-maximal distance from x into a hyperplane e∗(x) of Σ. Using a result of Kasikova and Shult [11], we are able the show that there exists up to isomorphisms a unique full polarized embedding of Δ of minimal dimension. We also show that e∗ realizes a full polarized embedding of Δ into a subspace of the dual of Σ, and that e∗ is isomorphic to the minimal full polarized embedding of Δ. In the final section, we will determine the minimal full polarized embeddings of the finite dual polar spaces DQ(2n,q), DQ ⁻(2n+1,q), DH(2n−1,q ²) and DW(2n−1,q) (q odd), but the latter only for n≤ 5. We shall prove that the minimal full polarized embeddings of DQ(2n,q), DQ ⁻(2n+1,q) and DH(2n−1,q ²) are the `natural' ones, whereas this is not always the case for DW(2n−1, q).B. De Bruyn: Postdoctoral Fellow of the Research Foundation - Flanders.     

Small weight codewords in the codes arising from Desarguesian projective planes

We study codewords of small weight in the codes arising from Desarguesian projective planes. We first of all improve the results of K. Chouinard on codewords of small weight in the codes arising from PG(2, p), p prime. Chouinard characterized all the codewords up to weight 2p in these codes. Using a particular basis for this code, described by Moorhouse, we characterize all the codewords of weight up to 2p + (p−1)/2 if p ≥ 11. We then study the codes arising from . In particular, for q ₀ = p prime, p ≥ 7, we prove that the codes have no codewords with weight in the interval [q + 2, 2q − 1]. Finally, for the codes of PG(2, q), q = p ^h , p prime, h ≥ 4, we present a discrete spectrum for the weights of codewords with weights in the interval [q + 2, 2q − 1]. In particular, we exclude all weights in the interval [3q/2, 2q − 1]. Geertrui Van de Voorde research is supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen) Joost Winne was supported by the Fund for Scientific Research - Flanders (Belgium).     

Complete Spans on Hermitian Varieties

Let L be a general linear complex in PG(3, q) for any prime power q. We show that when GF(q) is extended to GF(q ²), the extended lines of L cover a non-singular Hermitian surface H ? H(3, q ²) of PG(3, q ²). We prove that if Sis any symplectic spread PG(3, q), then the extended lines of this spread form a complete (q ² + 1)-span of H. Several other examples of complete spans of H for small values of q are also discussed. Finally, we discuss extensions to higher dimensions, showing in particular that a similar construction produces complete (q ³ + 1)-spans of the Hermitian variety H(5, q ²).     

Distribution of a certain partition function modulo powers of primes

In this paper, we study a certain partition function a(n) defined by Σ _n≥0 a(n)q ⁿ := Π _n=1(1 − q ⁿ )⁻¹(1 − q ²ⁿ )⁻¹. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m ^j ). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n).     

Relative cohomology of polynomial mappings

 Let F be a polynomial mapping from ℂ ⁿ to ℂ ^q with n>q. We study the De Rham cohomology of its fibres and its relative cohomology groups, by introducing a special fibre F ⁻¹(∞) ``at infinity' and its cohomology. Let us fix a weighted homogeneous degree on with strictly positive weights. The fibre at infinity is the zero set of the leading terms of the coordinate functions of F. We introduce the cohomology groups H ^k (F ⁻¹(∞)) of F at infinity. These groups enable us to compute all the other cohomology groups of F. For instance, if the fibre at infinity has an isolated singularity at the origin, we prove that every weighted homogeneous basis of H ^n−q (F ⁻¹ (∞)) is a basis of all the groups H ^n−q (F ⁻¹(y)) and also a basis of the (n−q) ^th relative cohomology group of F. Moreover the dimension of H ^n−q (F ⁻¹(∞)) is given by a global Milnor number of F, which only depends on the leading terms of the coordinate functions of F. Received: 12 February 2002 / Revised version: 25 May 2002 Published online: 3 March 2003     

On an Isomorphism Problem of Some Dual Hyperovals in PG(2d + 1, q) with q even

Let q = 2^l with l≥ 1 and d ≥ 2. We prove that any automorphism of the d-dimensional dual hyperoval over GF(q), constructed in [3] for any (d + 1)-dimensional GF(q)-vector subspace V in GF(qⁿ) with n≥ d + 1 and for any generator σ of the Galois group of GF(qⁿ) over GF(q), always fixes the special member X(∞). Moreover, we prove that, in case V = GF(q^d+1), two dual hyperovals and in PG(2d + 1,q), where σ and τ are generators of the Galois group of GF(q^d+1) over GF(q), are isomorphic if and only if (1) σ = τ or (2) σ τ = id. Therefore, we have proved that, even in the case q > 2, there exist non isomorphic d-dimensional dual hyperovals in PG(2d + 1,q) for d ≥ 3.     

Complete ( q 2?+ q +?8)/2-caps in the spaces PG (3, q ), q ≡ 2 (mod 3) an odd prime, and a complete 20-cap in PG (3, 5)

An infinite family of complete (q ² + q + 8)/2-caps is constructed in PG(3, q) where q is an odd prime ≡ 2 (mod 3), q ≥ 11. This yields a new lower bound on the second largest size of complete caps. A variant of our construction also produces one of the two previously known complete 20-caps in PG(3, 5). The associated code weight distribution and other combinatorial properties of the new (q ² + q + 8)/2-caps and the 20-cap in PG(3, 5) are investigated. The updated table of the known sizes of the complete caps in PG(3, q) is given. As a byproduct, we have found that the unique complete 14-arc in PG(2, 17) contains 10 points on a conic. Actually, this shows that an earlier general result dating back to the Seventies fails for q = 17.      

Symplectic embeddings of ellipsoids

We study the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid. It is first proved that under the conditionr _n ² ≤2r ₁ ² the symplectic ellipsoidE(r ₁,…,r _n)with radiir ₁≤…≤r _ndoes not symplectically embed into a ball of radius strictly smaller thanr _n.We then use symplectic folding to see that this condition is sharp. We finally sketch a proof of the fact that any connected symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids.