On Buekenhout-Metz unitals in PG(2,q 2),q even

Research partially supported by M.P.I. (Research project Strutture geometriche combinatorie e loro applicazioni).     

On codewords in the dual code of classical generalised quadrangles and classical polar spaces

In [J.L. Kim, K. Mellinger, L. Storme, Small weight codewords in LDPC codes defined by (dual) classical generalised quadrangles, Des. Codes Cryptogr. 42 (1) (2007) 73-92], the codewords of small weight in the dual code of the code of points and lines of Q(4,q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q⁺(5,q) and H(5,q²), and we present lower bounds on the weight of the codewords in the dual of the code of points and k-spaces of the classical polar spaces. Furthermore, we investigate the codewords with the largest weights in these codes, where for q even and k sufficiently small, we determine the maximum weight and characterise the codewords of maximum weight. Moreover, we show that there exists an interval such that for every even number w in this interval, there is a codeword in the dual code of Q⁺(5,q), q even, with weight w and we show that there is an empty interval in the weight distribution of the dual of the code of Q(4,q), q even. To prove this, we show that a blocking set of Q(4,q), q even, of size q²+1+r, where 0<r<(q+4)/6, contains an ovoid of Q(4,q), improving on [J. Eisfeld, L. Storme, T. Sz?nyi, P. Sziklai, Covers and blocking sets of classical generalised quadrangles, Discrete Math. 238 (2001) 35-51, Theorem 9].     

Small complete caps in PG(N,q), q even

A new family of small complete caps in PG(N,q), q even, is constructed. Apart from small values of either N or q, it provides an improvement on the currently known upper bounds on the size of the smallest complete cap in PG(N,q): for N even, the leading term is replaced by with , for N odd the leading term is replaced by with . © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 420–436, 2007     

New inductive constructions of complete caps in PG(N,q), q even

Some new families of small complete caps in PG(N, q), q even, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem in the plane. The caps constructed in this article provide an improvement on the currently known upper bounds on the size of the smallest complete cap in PG(N, q), N≥4, for all q≥2³. In particular, substantial improvements are obtained for infinite values of q square, including q=2^2Cm, C≥5, m≥3; for q=2^Cm, C≥5, m≥9, with C, m odd; and for all q≤2¹⁸. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 177–201, 2010     

Blocking sets of nonsecant lines to a conic in PG(2,q), q odd

In a previous paper 1 , all point sets of minimum size in PG(2,q), blocking all external lines to a given irreducible conic , have been determined for every odd q. Here we obtain a similar classification for those point sets of minimum size, which meet every external and tangent line to . © 2004 Wiley Periodicals, Inc.     

Large blocking sets in PG(2,q2)

Minimal blocking sets in $\mathrm{PG}(2,q^2)$ have size at most $q^3+1$. This result is due to Bruen and Thas and the bound is sharp, sets attaining this bound are called unitals. In this paper, we show that the second largest minimal blocking sets have size at most $q^3+1-(p-3)/2$, if $q=p$, $p\ge 67$, or $q=p^h$, $p>7$, $h>1$. Our proof also works for sets having at least one tangent at each of its points (that is, for tangency sets).     

On antipodal Euclidean tight (2e + 1)-designs

Neumaier and Seidel (1988) generalized the concept of spherical designs and defined Euclidean designs in ℝ ⁿ . For an integer t, a finite subset X of ℝ ⁿ given together with a weight function w is a Euclidean t-design if holds for any polynomial f(x) of deg(f)≤ t, where {S _i , 1≤ i ≤ p} is the set of all the concentric spheres centered at the origin that intersect with X, X _i = X∩ S _i , and w:X→ ℝ_{> 0}. (The case of X⊂ S ⁿ⁻¹ with w≡ 1 on X corresponds to a spherical t-design.) In this paper we study antipodal Euclidean (2e+1)-designs. We give some new examples of antipodal Euclidean tight 5-designs. We also give the classification of all antipodal Euclidean tight 3-designs, the classification of antipodal Euclidean tight 5-designs supported by 2 concentric spheres.     

A characterization of Buekenhout-Metz unitals in PG(2, q 2), q even

The known examples of embedded unitals (i.e. Hermitian arcs) in PG(2, q ²) are B-unitals, i.e. they can be obtained from ovoids of PG(3, q) by a method due to Buekenhout. B-unitals arising from elliptic quadrics are called BM-unitals. Recently, BM-unitals have been classified and their collineation groups have been investigated. A new characterization is given in this paper. We also compute the linear collineation group fixing the B-unital arising from the Segre-Tits ovoid of PG(3, 2 ^r ), r3 odd. It turns out that this group is an Abelian group of order q ².Research supported by MURST.     

On the number of pairwise non-isomorphic minimal blocking sets in PG(2, q)

In this paper we examine whether the number of pairwise non-isomorphic minimal blocking sets in PG(2, q) of a certain size is larger than polynomial. Our main result is that there are more than polynomial pairwise non-isomorphic minimal blocking sets for any size in the intervals [2q−1, 3q−4] for q odd and for q square. We can also prove a similar result for certain values of the intervals and .      

A condition for the existence of ovals in PG(2, q), q even

A condition is found that determines whether a polynomial over GF(q) gives an oval in PG(2, q), q even. This shows that the set of all ovals of PG(2, q) corresponds to a certain variety of points of PG((q–4)/2, q). The condition improves upon that of Segre and Bartocci, who proved that all the terms of an oval polynomial have even degree. It is suitable for efficient computer searches.     

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

On the structure of generalized symmetric spaces of SLn𝔽q)

In this paper we extend previous results regarding SL₂(k) over any finite field k by investigating the structure of the symmetric spaces for the family of special linear groups SL_n(k) for any integer n>2. Specifically, we discuss the generalized and extended symmetric spaces of SL_n(k) for all conjugacy classes of involutions over a finite field of odd or even characteristic. We characterize the structure of these spaces and provide an explicit difference set in cases where the two spaces are not equal.     

On the quasi-regularity of non-sectorial Dirichlet forms by processes having the same polar sets

We obtain a criterion for the quasi-regularity of generalized (non-sectorial) Dirichlet forms, which extends the result of P.J. Fitzsimmons on the quasi-regularity of (sectorial) semi-Dirichlet forms. Given the right (Markov) process associated to a semi-Dirichlet form, we present sufficient conditions for a second right process to be a standard one, having the same state space. The above mentioned quasi-regularity criterion is then an application. The conditions are expressed in terms of the associated capacities, nests of compacts, polar sets, and quasi-continuity. The second application is on the quasi-regularity of the generalized Dirichlet forms obtained by perturbing a semi-Dirichlet form with kernels.