Spreads In PG Research Articles

On parallelisms of PG(3,4) with automorphisms of order 2

Let PG(n,q) be the n-dimensional projective space over the finite field Fq. A spread in PG(n,q) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, and cryptography. All parallelisms of PG(3,2) and PG(3,3) are known. Parallelisms of PG(3,4) which are invariant under automorphisms of odd prime orders have also been classified. The present paper contributes to the classification of parallelisms in PG(3,4) with automorphisms of even order. We focus on cyclic groups of order four and the group of order two generated by a Baer involution. We examine invariants of the parallelisms such as the full automorphism group, the type of spreads and questions of duality. The results given in this paper show that the number of parallelisms of PG(3,4) is at least 8675365. Some future directions of research are outlined.

Transitive Deficiency One Parallelisms of PG(3, 7)

Consider the n-dimensional projective space PG(n,q) over a finite field with q elements. A spread in PG(n,q) is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. A deficiency one parallelism is a partial parallelism with one spread less than the parallelism. A transitive deficiency one parallelism corresponds to a parallelism possessing an automorphism group which fixes one spread and is transitive on the remaining spreads. Such parallelisms have been considered in many papers. As a result, an infinite family of transitive deficiency one parallelisms of PG(n,q) has been constructed for odd q, and it has been proved that the deficiency spread of a transitive deficiency one parallelism must be regular, and its automorphism group should contain an elation subgroup of order q2. In the present paper we construct parallelisms of PG(3,7) invariant under an elation group of order 49 with some additional properties, and thus we succeed to obtain all (46) transitive deficiency one parallelisms of PG(3,7). The three parallelisms from the known infinite family are among them. As a by-product, we also construct a much bigger number (55,022) of parallelisms which have the same spread structure, but are not transitive deficiency one.

New Uniform Subregular Parallelisms of PG(3, 4) Invariant under an Automorphism of Order 2

Abstract A spread in PG(n, q) is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. A parallelism is uniform if all its spreads are isomorphic. Up to isomorphism, there are three spreads of PG(3, 4) – regular, subregular and aregular. Therefore, three types of uniform parallelisms are possible. In this work, we consider uniform parallelisms of PG(3, 4) which possess an automorphism of order 2. We establish that there are no regular parallelisms, and that there are 8253 nonisomorphic subregular parallelisms. Together with the parallelisms known before this work, this yields a total of 8623 known subregular parallelisms of PG(3, 4).

On symplectic semifield spreads of PG(5,q 2), q odd

Abstract We prove that there exist exactly three non-equivalent symplectic semifield spreads of PG ⁡ ( 5 , q 2 ) {\operatorname{PG}(5,q^{2})} , for q 2 > 2 ⋅ 3 8 {q^{2}>2\cdot 3^{8}} odd, whose associated semifield has center containing 𝔽 q {\mathbb{F}_{q}} . Equivalently, we classify, up to isotopy, commutative semifields of order q 6 {q^{6}} , for q 2 > 2 ⋅ 3 8 {q^{2}>2\cdot 3^{8}} odd, with middle nucleus containing 𝔽 q 2 {\mathbb{F}_{q^{2}}} and center containing 𝔽 q {\mathbb{F}_{q}} .

The maximum size of a partial spread in a finite projective space

Let n and t be positive integers with t<n, and let q be a prime power. A partial(t−1)-spread of PG(n−1,q) is a set of (t−1)-dimensional subspaces of PG(n−1,q) that are pairwise disjoint. Let r=n mod t and 0≤r<t. We prove that if t>(qr−1)/(q−1), then the maximum size, i.e., cardinality, of a partial (t−1)-spread of PG(n−1,q) is (qn−qt+r)/(qt−1)+1. This essentially settles a main open problem in this area. Prior to this result, this maximum size was only known for r∈{0,1} and for r=q=2.

The maximum size of a partial spread II: Upper bounds

Let n and t be positive integers with t<n, and let q be a prime power. A partial(t−1)-spread of PG(n−1,q) is a set of (t−1)-dimensional subspaces of PG(n−1,q) that are pairwise disjoint. Let r≡n(modt) with 0≤r<t, and let Θi=(qi−1)∕(q−1). We essentially prove that if 2≤r<t≤Θr, then the maximum size of a partial (t−1)-spread of PG(n−1,q) is bounded from above by (Θn−Θt+r)∕Θt+qr−(q−1)(t−3)+1. We actually give tighter bounds when certain divisibility conditions are satisfied. These bounds improve on the previously known upper bound for the maximum size partial (t−1)-spreads of PG(n−1,q); for instance, when ⌈Θr2⌉+4≤t≤Θr and q>2. The exact value of the maximum size partial (t−1)-spread has been recently determined for t>Θr by the authors of this paper (see Năstase and Sissokho [21] (2016)).

On point-transitive and transitive deficiency one parallelisms of PG $$(3,4)$$ ( 3 , 4 )

A parallelism in PG $$(n,q)$$ ( n , q ) is point-transitive if it has an automorphism group which is transitive on the points. If the automorphism group fixes one spread and is transitive on the remaining spreads, the parallelism corresponds to a transitive deficiency one parallelism. It is known that there are three types of spreads in PG $$(3,4)$$ ( 3 , 4 ) --regular, subregular and aregular. A parallelism is regular if all its spreads are regular. In PG $$(3,4)$$ ( 3 , 4 ) no point-transitive parallelisms, no regular ones, and no transitive deficiency one parallelisms have been known. Both point-transitive parallelisms and transitive deficiency one parallelisms must have automorphisms of order 5. We construct all 32,048 nonisomorphic parallelisms with automorphisms of order 5 and classify them by the orders of their automorphism groups and by the types of their spreads. There are 31,832 parallelisms with an automorphism group fixing exactly one spread. Only for four of them the automorphism group is transitive on the remaining spreads. Among the parallelisms we construct there are no regular ones. There are 4,124 parallelisms with automorphisms of order 5 without fixed points, but none of them is point-transitive.

Families of twisted tensor product codes

Using geometric properties of the variety $${\mathcal V_{r,t}}$$ , the image under the Grassmannian map of a Desarguesian (t − 1)-spread of PG(rt − 1, q), we introduce error correcting codes related to the twisted tensor product construction, producing several families of constacyclic codes. We determine the precise parameters of these codes and characterise the words of minimum weight.

Trivectors yielding spreads in PG(5, 2)

Given an alternating trilinear form $${T\in {\rm Alt}(\times^{3}V_{6})}$$ on V 6 = V(6, 2) let $${\mathcal{L}_{T}}$$ denote the set of those lines $${\langle a, b \rangle}$$ in $${{\rm PG}(5,2)=\mathbb{P}V_{6}}$$ which are T-singular, satisfying, that is, T(a, b, x) = 0 for all $${x\in {\rm PG}(5, 2).}$$ If $${\mathcal{L}_{21}}$$ is a Desarguesian line-spread in PG(5, 2) it is shown that $${\mathcal{L}_{T}=\mathcal{L}_{21}}$$ for precisely three choices T 1,T 2,T 3 of T, which moreover satisfy T 1 + T 2 + T 3 = 0. For $${T\in\mathcal{T}:=\{T_{1},T_{2},T_{3}\}}$$ the $${\mathcal{G}_{T}}$$ -orbits of flats in PG(5, 2) are determined, where $${\mathcal{G}_{T}\cong {\rm SL}(3,4).2}$$ denotes the stabilizer of T under the action of GL(6, 2). Further, for a representative U of each $${\mathcal{G}_{T}}$$ -orbit, the T-associate U # is also determined, where by definition $$U^{\#}=\{v\in {\rm PG}(5,2)\, |\, T(u_{1},u_{2},v) = 0\, \,{\rm for\,all }\, \, u_{1},u_{2}\in U\}$$ .

A remark on symplectic spreads

We first note that each element of a symplectic spread of PG(2n − 1, 2r) either intersects a suitable nonsingular quadric in a subspace of dimension n − 2 or is contained in it, then we prove that this property characterises symplectic spreads of PG(2n − 1, 2r). As an application, we show that a translation plane of order qn, q even, with kernel containing GF(q), is defined by a symplectic spread if and only if it contains a maximal arc of the type constructed by Thas (Europ J Combin 1:189–192, 1980).

Lifting Subregular Spreads

Let S k denote a set of k reguli in a Desarguesian affine plane $$\sum_{q^2}$$ of order q 2. It is shown that, for every odd integer s > 1, there is a corresponding set S s k of k reguli in any Desarguesian plane $$\sum_{q^{2s}}$$ of order q 2s such the line intersection properties of the reguli of S s k are inherited from those of S k . Hence, sets of mutually disjoint reguli in $$\sum_{q^2}$$ ‘lift’ to sets of mutually disjoint reguli in $$\sum_{q^{2s}}$$ . Thus, the existence of a subregular spread in PG(3, q) produces an infinite class of subregular spreads in spaces PG(3, q s ).

Random constructions and density results

In this paper we outline a construction method which has been used for minimal blocking sets in PG(2, q) and maximal partial line spreads in PG(n, q) and which must have a lot of more applications. We also give a survey on what is known about the spectrum of sizes of maximal partial line spreads in PG(n, q). At the end we list some more elaborate random techniques used in finite geometry.

On affine designs and Hadamard designs with line spreads

Rahilly [On the line structure of designs, Discrete Math. 92 (1991) 291–303] described a construction that relates any Hadamard design H on 4 m - 1 points with a line spread to an affine design having the same parameters as the classical design of points and hyperplanes in AG ( m , 4 ) . Here it is proved that the affine design is the classical design of points and hyperplanes in AG ( m , 4 ) if, and only if, H is the classical design of points and hyperplanes in PG ( 2 m - 1 , 2 ) and the line spread is of a special type. Computational results about line spreads in PG ( 5 , 2 ) are given. One of the affine designs obtained has the same 2-rank as the design of points and planes in AG ( 3 , 4 ) , and provides a counter-example to a conjecture of Hamada [On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes, Hiroshima Math. J. 3 (1973) 153–226].

Mixed partitions and related designs

We define a mixed partition of ? = PG(d, q r ) to be a partition of the points of ? into subspaces of two distinct types; for instance, a partition of PG(2n ? 1, q 2) into (n ? 1)-spaces and Baer subspaces of dimension 2n ? 1. In this paper, we provide a group theoretic method for constructing a robust class of such partitions. It is known that a mixed partition of PG(2n ? 1, q 2) can be used to construct a (2n ? 1)-spread of PG(4n ? 1, q) and, hence, a translation plane of order q 2n . Here we show that our partitions can be used to construct generalized Andre planes, thereby providing a geometric representation of an infinite family of generalized Andre planes. The results are then extended to produce mixed partitions of PG(rn ? 1, q r ) for r ? 3, which lift to (rn ? 1)-spreads of PG(r 2 n ? 1, q) and hence produce $$2-(q^{r^2n},q^{rn},1)$$ (translation) designs with parallelism. These designs are not isomorphic to the designs obtained from the points and lines of AG(r, q rn ).

Spreads of PG(3, q) and ovoids of polar spaces

To any spread 𝒮 of PG (3, q ) corresponds a family of locally hermitian ovoids of the Hermitian surface H (3, q 2 ), and conversely; if in addition 𝒮 is a semifield spread, then each associated ovoid is a translation ovoid, and conversely. In this paper we calculate the translation group of the locally hermitian ovoids of H (3, q 2 ) arising from a given semifield spread, and we characterize the p-semiclassical ovoid constructed in [Cossidente A., Ebert G., Marino G., Siciliano A.: Shult Sets and Translation Ovoids of the Hermitian Surface. Adv. Geom. 6 (2006), 475–494] as the only translation ovoid of H (3, q 2 ) whose translation group is abelian. If 𝒮 is a spread of PG (3, q ) and 𝒪(𝒮) is one of the associated ovoids of H (3, q 2 ), then using the duality between H (3, q 2 ) and Q − (5, q ), another spread of PG (3, q ), say 𝒮 2 , can be constructed. On the other hand, using the Barlotti-Cofman representation of H (3, q 2 ), one more spread of a 3-dimensional projective space, say 𝒮 1 , arises from the ovoid 𝒪(𝒮). In [Lunardon G.: Blocking sets and semifields. J. Comb. Theory Ser. A to appear] some questions are posed on the relations among 𝒮, 𝒮 1 and 𝒮 2 ; here we prove that 𝒮 and 𝒮 2 are isomorphic and the ovoids 𝒪(𝒮) and 𝒪(𝒮 1 ), corresponding to 𝒮 and 𝒮 1 respectively under the Plücker map, are isomorphic.

Skew Hadamard difference sets from the Ree–Tits slice symplectic spreads in [formula omitted

Using a class of permutation polynomials of F 3 2 h + 1 obtained from the Ree–Tits slice symplectic spreads in PG ( 3 , 3 2 h + 1 ) , we construct a family of skew Hadamard difference sets in the additive group of F 3 2 h + 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.

Collineation groups of translation planes admitting hyperbolic Buekenhout or parabolic Buekenhout–Metz unitals

It is shown that for every semifield spread in PG ( 3 , q ) and for every parabolic Buekenhout–Metz unital, there is a collineation group of the associated translation plane that acts transitively and regularly on the affine points of the parabolic unital. Conversely, any spread admitting such a group is shown to be a semifield spread. For hyperbolic Buekenhout unitals, various collineation groups of translation planes admitting such unitals and the associated planes are determined.

Transitive parabolic unitals in semifield planes

Every semifield plane with spread in PG(3,K), where K is a field admitting a quadratic extension K+, is shown to admit a transitive parabolic unital.

Subregular Spreads of Hermitian Unitals

In this paper, we consider the problem of constructing partitions of the points of a Hermitian unital into pairwise disjoint blocks, commonly known as spreads. We generalize a construction of Baker et al. (In Finite Geometry and Combinatorics, Vol. 191 of London Math. Soc. Lecture Not Ser., pages 17---30. Cambridge University Press, Cambridge, 1993.) to provide a new infinite family of spreads. Morover, we develop a structural connection between these new spreads of the Hermitian unital in PG(2, q2) and the subregular spreads of PG(3, q), allowing us to christen a new "subregular" family of spreads in the Hermitian unital in PG(2, q2).

Partial spreads in PG(4,2) and flats in PG(9,2) external to the Grassmannian [formula omitted

We consider the following ‘ even hyperplane construction’ of flats in the projective space PG ( 9 , 2 ) = P ( ∧ 2 V ( 5 , 2 ) ) which are external to the Grassmannian G 1 , 4 , 2 of lines of PG(4,2). Let the Grassmann image in G 1 , 4 , 2 of a partial spread S r = { μ 1 , … , μ r } in PG(4,2) be C r = { m 1 , … , m r } . Then C r is an r-cap on G 1 , 4 , 2 . Using the recent classification [N.A. Gordon, R. Shaw, L.H. Soicher, Classification of Partial Spreads in PG(4,2), pp. 63, available from: http://www.hull.ac.uk/maths/people/rs/staffdetails.html] of partial spreads in PG(4,2), we determine those partial spreads S r such that the projective space E ( C r ) = 〈 C r 〉 even generated by the r 2 points m ij ≔ m i + m j is an external flat. We show that, in this simple manner, we may construct seven out of the ten GL(5,2)-orbits of external flats.