PD-sets for the codes related to some classical varieties

We generalize the notion of a PD-set of a code to that of a t-PD-set of an arbitrary permutation set. We find PD-sets for miquelian Benz planes of small order and for the ruled rational normal surface of order 3 in PG(4,3) and in PG(4,4). These results yield PD-sets for the related linear codes.     

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.     

From primary to dual affine variety codes over the Klein quartic

Designs, Codes and Cryptography - In Geil and Özbudak (Cryptogr Commun 11(2):237–257, 2019) a novel method was established to estimate the minimum distance of primary affine variety...     

Clifford parallelisms and external planes to the Klein quadric

For any three-dimensional projective space \({\mathbb{P}(V)}\), where V is a vector space over a field F of arbitrary characteristic, we establish a one-one correspondence between the Clifford parallelisms of \({\mathbb{P}(V)}\) and those planes of \({\mathbb{P} (V \wedge V)}\) that are external to the Klein quadric representing the lines of \({\mathbb{P}(V)}\). We also give two characterisations of a Clifford parallelism of \({\mathbb{P}(V)}\), both of which avoid the ambient space of the Klein quadric.     

On Point-Cyclic Resolutions of the 2-(63,7,15) Design Associated with PG(5,2)

 A t(v,k,λ) design is a set of v points together with a collection of its k-subsets called blocks so that t points are contained in exactly λ blocks. PG(n,q), the n-dimensional projective geometry over GF(q) is a 2(q ⁿ +q ⁿ⁻¹ +⋯+q+1,q ²+q+1, q ⁿ⁻² + q ⁿ⁻³ +⋯+q+1) design when we take its points as the points of the design and its planes as the blocks of the design. A 2(v,k,λ) design is said to be resolvable if the blocks can be partitioned as ℱ={R ₁,R ₂,…,R _s }, where s=λ(v−1)/(k−1) and each R _i consists of v/k disjoint blocks. If a resolvable design has an automorphism σ which acts as a cycle of length v on the points and ℱ^σ=ℱ, then the design is said to be point-cyclically resolvable. The design consisting of points and planes of PG(5,2) is shown to be point-cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G=〈σ〉 where σ is a cycle of length v. These resolutions are shown to be the only resolutions which admit point-transitive automorphism group. Received: November 10, 1999 Final version received: September 18, 2000 Acknowledgments. The author would like to thank A. Munemasa for his assistance in writing computer programs on constructing projective spaces and searching for partial spreads. Moreover, she's thankful to T. Hishida and M.␣Jimbo for helpful discussions and for verifying the results of this paper. Present address: Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City 1108, Philippines. e-mail: jumela@mathsci.math.admu.edu.ph     

A combinatorial characterization of the Klein quadric

We give a combinatorial characterization of the Klein quadric in terms of its incidence structure of points and lines. As an application, we obtain a combinatorial proof of a result of Havlicek.In memoriam Giuseppe TalliniWork supported by National Research Project Strutture Geometriche, Combinatoria e loro applicazioni of the Italian Ministere dell'Università e della Ricerca Scientifica and by G.N.S.A.G.A. of C.N.R.      

A6-invariant ovoids of the Klein quadric

Translation planes associated with A₆-invariant ovoids of the Klein quadric are discussed.     

Notes on binary codes related to the O(5, q) generalized quadrangle for odd q

In this note we determine the dimensions of the binary codes spanned by the lines or by the point neighborhoods in the generalized quadrangle Sp(4, q) and its dual O(5, q), where q is odd. Several more general results are given. As a side result we find that if a square generalized quadrangle of odd order has an antiregular point, then all of its points are antiregular.On leave from the Indian Statistical Institute, Calcutta; research supported by a grant from NWO.     

A characterization of the family of lines external to a hyperbolic quadric of PG (3, q)

In this paper we characterize the family of external lines to a hyperbolic quadric of PG (3, q).     

Optimal pseudo-cyclic codes and caps in PG (3,q)

Existence and uniqueness of pseudo-cyclic [q ²+1,q ²–3, 4]-codes over GF(q) are proved. Elliptic quadrics are characterized as those (q ²+1)-caps in PG(3,q) whose corresponding [q ²+1,q ²–3, 4]-codes are pseudo-cyclic.     

The cubic Segre variety in PG(5, 2)

The Segre variety in PG(5, 2) is a 21-set of points which is shown to have a cubic equation Q(x) = 0. If T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing the cubic polynomial Q, then the associate U ^# of an r-flat is defined to be

Flags in affine planes and maximal spreads in a non-singular quadric of PG(4,q)

We give an example of maximal spreadF in a non-singular quadric of PG(4,q), with ¦F¦ = 3q + 1.     

A characterisation of the planes meeting a non-singular quadric of PG(4,Q) in a conic

By counting and geometric arguments, we provide a combinatorial characterisation of the planes meeting the non-singular quadric of PG(4,q) in a conic. A characterisation of the tangents and generators of this quadric when q is odd has been proved by de Resmini [15], and we give an alternative using our result.     

Stabilizers of subspaces under similitudes of the Klein quadric,and automorphisms of Heisenberg algebras

We determine the groups of automorphisms and their orbits for nilpotent Lie algebras of class 2 and small dimension, over arbitrary fields (including the characteristic 2 case).