On (q 2 + q + 1)-sets of class [1, m,n]2 in PG(3, q)

A characterization of cones in PG(3, q) as sets of points of PG(3, q) of size q ² + q + 1 projecting from a point V a set of q + 1 points of a plane of PG(3, q) and with three intersection numbers with respect to the planes is given.     

42-arcs in PG(2, q) left invariant by PSL(2, 7)

For an odd prime p?≠ 7, let q be a power of p such that ${q^3\equiv1 \pmod 7}$ . It is known that the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to PSL(2, 7). For such a projective group Γ, we investigate the geometric properties of the (unique) Γ-orbit Ω of size 42 such that the 1-point stabilizer of Γ in Ω is a cyclic group of order 4. We present a computational approach to prove that Ω is a 42-arc provided that q?≥ 53 and q?≠ 37³, 11⁶, 5⁶, 3⁶. We discuss the case q?=?53 in more detail showing the completeness of Ω for q?=?53.     

Cyclic Arcs in PG(2, q)

B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete (q ² ? q + l)-arcs in PG(2, q ²), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q ² ? q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.     

An infinite family of type (m,n) sets in PG(2,q2), q a square

A k-set of type (m,n), with k=(q+√q+1)(q²?q+1), m= 1+√q, n=q+√q+1, is proved to exist in a Galois plane PG(2,q²), q a square, and its construction is given. Thus, its complement, i.e. a ((q?√q)(q+√q+1)(q²?q+1); √q(q√q?√q?1),√q(q √q?1))-set, exists too. The special case q=16 is considered and the points of a (91;3,7)-set in PG(2,16) are exhibited. A generalization is given.     

On sets of type (q + 1, n)2 in finite three-dimensional projective spaces

It is proved that a k–set of type (q + 1, n)₂ in PG(3, q) either is a plane or it has size k ≥ (q + 1)² and a characterization of some sets of size (q + 1)² is given.     

Linear (q+1)-fold Blocking Sets in PG(2, q4)

A (q+1)-fold blocking set of size (q+1)(q⁴+q²+1) in PG(2, q⁴) which is not the union of q+1 disjoint Baer subplanes, is constructed     

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

More than thirty new upper bounds on the smallest size t ₂(2, q) of a complete arc in the plane PG(2, q) are obtained for (169 ≤ q ≤ 839. New upper bounds on the smallest size t ₂(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m ₂(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m ₂(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t ₂(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 ^h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete ${(\frac{1}{2}(q + 3) + \delta)}$ -arcs other than conics that share ${\frac{1}{2}(q + 3)}$ points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), ${{q \equiv 2}}$ (mod 3) odd, we propose new constructions of ${\frac{1}{2} (q + 7)}$ -arcs and show that they are complete for q ≤ 3701.     

The existence of maximal (q 2, 2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic

We prove that (q ², 2)-arcs exist in the projective Hjelmslev plane PHG(2, R) over a chain ring R of length 2, order |R| = q ² and prime characteristic. For odd prime characteristic, our construction solves the maximal arc problem. For characteristic 2, an extension of the above construction yields the lower bound q ² + 2 on the maximum size of a 2-arc in PHG(2, R). Translating the arcs into codes, we get linear [q ³, 6, q ³ ?q ² ?q] codes over ${\mathbb {F}_q}$ for every prime power q > 1 and linear [q ³ + q, 6,q ³ ?q ² ?1] codes over ${\mathbb {F}_q}$ for the special case q = 2 ^r . Furthermore, we construct 2-arcs of size (q + 1)²/4 in the planes PHG(2, R) over Galois rings R of length 2 and odd characteristic p ².     

Skew-Hadamard matrices of order 2(q + 1)

Szekeres has established the existence of a skew-Hadamard matrix of order 2(q + 1) in the case q ≡ 5 (mod 8), a prime power. His method utilized complementary difference sets in the elementary abelian group of order q. The main result of this paper is to show that, for the same q, there exist skew-Hadamard matrices of order 2(q + 1) that are of the Goethals-Seidel type. This is achieved by using a cyclic relative difference set with parameters $\text{(q + 1, 4, q,}\text{1}\text{4}\text{(q ? 1))}$.     

An integral transform in L2-cohomology for the ladder representations of U(p, q)

Starting from the realization of the Fock space as L²-cohomology of $\text{C}$^{p + q}, $\text{H}$^0,p($\text{C}$^{p + q}) = ⊕_m?$\text{Z}$$\text{H}$_m^0,p($\text{C}$^{p + q}), an integral transform is constructed which is a direct-image mapping from $\text{H}$_m^{0,p($\text{C}$p + q) into the space of holomorphic sections of some vector bundle E_m over M ≈ U(p, q)/(U(q) × U(p)), m ? 0}. The transform intertwines the natural actions of U(p, q) and is injective if m ? 0, so it provides a geometric realization of the ladder representations of U(p, q). The sections in the image of the transform satisfy certain linear differential equations, which are explicitly described. For example, Maxwell's equations are of this form if p = q = 2 and m = 2. Thus, this transform is analogous to the Penrose correspondence.     

On subgroups and factor groups of GL(n, q) acting on spreads with the same characteristic

Let π^l_∞ be an affine translation plane of order q^r with GF(q) in its kern. Suppose G is a subgroup of the translation complement of π^l_∞ which leaves invariant a set Δ of q + 1 slopes and acts transitively on l_∞?Δ. We study the situation when G≌SL(n, q) or PSL(n, q).We show that if G|Δ = identity, then π^l_∞ is a Hall plane, a Lorimer-Rahilly plane (LR-16) or a Johnson-Walker plane (JW-16). Moreover, if n?3, then G fixes Δ elementwise and π^l_∞ is LR-16 or JW-16.     

On translation planes of orderq 2 that admit a group of orderq 2 (q−1); bartolone's theorem

This article is concerned with translation planesP of orderq ² and kernelK isomorphic toG F(q). IfP admits a collineation groupG in the linear translation. complement and the order ofG K/K isq ²(q?1) then it is shown thatP is either a semifield plane or is a Lüneburg-Tits, Walker or Betten plane. This generalizes earlier work of Bartolone.     

Archi completi di ordine (q+3)/2 nei piani di galois,S 2,q,conq≡3 (mod. 4)

In a Galois planeS _2,q (q≡3 (mod 4),q>11) complete arcs of order (q+3)/2 are constructed.     

Orthogonal resolutions of lines in AG(n,q)

A resolution of the lines of AG(n,q) is a partition of the lines classes (called resolution classes) such that every point of the geometry is on exactly one line of each resolution class. Two resolutions R,R' of AG(n,q) are orthogonal if any resolution class from R has at most one line in common with any class from R'. In this paper, we construct orthogonal resolutions on AG(n,q) for all n=2ⁱ⁺¹, i=1,2,…, and all q>2 a prime power. The method involves constructing AG(n,q) from a finite projective plane of order q^n-1 and using the structure of the plane to display the orthogonal resolutions.     

Transitive A 6-invariant k-arcs in PG(2, q)

For q = p ^r with a prime p ≥ 7 such that ${q \equiv 1}$ or 19 (mod 30), the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to the alternating group A ₆ of degree 6. For a projectivity group ${\Gamma \cong A_6}$ of PG(2, q), we investigate the geometric properties of the (unique) Γ-orbit ${\mathcal{O}}$ of size 90 such that the 1-point stabilizer of Γ in its action on ${\mathcal O}$ is a cyclic group of order 4. Here ${\mathcal O}$ lies either in PG(2, q) or in PG(2, q ²) according as 3 is a square or a non-square element in GF(q). We show that if q ≥ 349 and q ≠ 421, then ${\mathcal O}$ is a 90-arc, which turns out to be complete for q = 349, 409, 529, 601,661. Interestingly, ${\mathcal O}$ is the smallest known complete arc in PG(2,601) and in PG(2,661). Computations are carried out by MAGMA.     

On the existence of Hadamard matrices

Given any natural number q > 3 we show there exists an integer t ? [2log₂(q ? 3)] such that an Hadamard matrix exists for every order 2^sq where s > t. The Hadamard conjecture is that s = 2.This means that for each q there is a finite number of orders 2^υq for which an Hadamard matrix is not known. This is the first time such a statement could be made for arbitrary q.In particular it is already known that an Hadamard matrix exists for each 2^sq where if q = 2^m ? 1 then s ? m, if q = 2^m + 3 (a prime power) then s ? m, if q = 2^m + 1 (a prime power) then s ? m + 1.It is also shown that all orthogonal designs of types (a, b, m ? a ? b) and (a, b), 0 ? a + b ? m, exist in orders m = 2^t and 2^t+2 · 3, t ? 1 a positive integer.     

On a set of lines of PG(3, q) corresponding to a maximal cap contained in the Klein quadric of PG (5, q)

The problem is considered of constructing a maximal set of lines, with no three in a pencil, in the finite projective geometry PG(3, q) of three dimensions over GF(q). (A pencil is the set of q+1 lines in a plane and passing through a point.) It is found that an orbit of lines of a Singer cycle of PG(3, q) gives a set of size q ³ + q ² + q + 1 which is definitely maximal in the case of q odd. A (q ³ + q ² + q + 1)-cap contained in the hyperbolic (or Klein) quadric of PG(5, q) also comes from the construction. (A k-cap is a set of k points with no three in a line.) This is generalized to give direct constructions of caps in quadrics in PG(5, q). For q odd and greater than 3 these appear to be the largest caps known in PG(5, q). In particular it is shown how to construct directly a large cap contained in the Klein quadric, given an ovoid skew to an elliptic quadric of PG(3, q). Sometimes the cap is also contained in an elliptic quadric of PG(5, q) and this leads to a set of q ³ + q ² + q + 1 lines of PG(3,q ²) contained in the non-singular Hermitian surface such that no three lines pass through a point. These constructions can often be applied to real and complex spaces.     

Blocking Sets in PG(r, q n )

Let $\mathcal S$ be a Desarguesian (n – 1)-spread of a hyperplane Σ of PG(rn, q). Let Ω and ${\bar B}$ be, respectively, an (n – 2)-dimensional subspace of an element of $\mathcal S $ and a minimal blocking set of an ((r – 1)n + 1)-dimensional subspace of PG(rn, q) skew to Ω. Denote by K the cone with vertex Ω and base ${\bar B}$ , and consider the point set B defined by $$B=\left(K\setminus\Sigma\right)\cup \{X\in \mathcal S\, : \, X\cap K\neq \emptyset\}$$ in the Barlotti–Cofman representation of PG(r, q ⁿ ) in PG(rn, q) associated to the (n – 1)-spread $\mathcal S$ . Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61–81, 2006), under suitable assumptions on ${\bar B}$ , we prove that B is a minimal blocking set in PG(r, q ⁿ ). In this way, we achieve new classes of minimal blocking sets and we find new sizes of minimal blocking sets in finite projective spaces of non-prime order. In particular, for q a power of 3, we exhibit examples of r-dimensional minimal blocking sets of size q ⁿ⁺² + 1 in PG(r, q ⁿ ), 3 ≤ r ≤ 6 and n ≥ 3, and of size q ⁴ + 1 in PG(r, q ²), 4 ≤ r ≤ 6; actually, in the second case, these blocking sets turn out to be the union of q ³ Baer sublines through a point. Moreover, for q an even power of 3, we construct examples of minimal blocking sets of PG(4, q) of size at least q ² + 2. From these constructions, we also get maximal partial ovoids of the hermitian variety H(4, q ²) of size q ⁴ + 1, for any q a power of 3.     

Regular Hadamard Matrices Generating Infinite Families of Symmetric Designs

If H is a regular Hadamard matrix with row sum 2h, m is a positive integer, and q = (2h ? 1)², then (4h ²(q ^{m + 1} ? 1)/(q ?1),(2h ² ? h)q ^m ,(h ²-h)q ^m ) are feasible parameters of a symmetric designs. If q is a prime power, then a balanced generalized weighing matrix BGW((q ^{m +1} ? 1)/(q?1),q ^m ,q ^m ?q ^{m ?1}) can be applied to construct such a design if H satisfies certain structural conditions. We describe such conditions and show that if H satisfies these conditions and B is a regular Hadamard matrix of Bush type, then B×H satisfies these structural conditions. This allows us to construct parametrically new infinite families of symmetric designs.     

Actions of GLq (2,C) on C(1,3) and its four dimensional representations

A complete classification is given of all inner actions on the Clifford algebra C(l,3) defined by representations of the quantum group GL_q (2,C)q^m ≠1, which are not reduced to representations of two commuting “q-spinors”. As a consequence of this classification it is shown that the space of invariants of every GL_q (2,C)-action of this type, which is not an action of SL_q (2,C), is generatedby 1 and the value of the quantum determinant for the given representation.