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.     

Baer subgeometry partitions

For any odd integern 3 and prime powerq, it is known thatPG(n–1, q²) can be partitioned into pairwise disjoint subgeometries isomorphic toPG(n–1, q) by taking point orbits under an appropriate subgroup of a Singer cycle ofPG(n–1, q²). In this paper, we construct Baer subgeometry partitions of these spaces which do not arise in the classical manner. We further illustrate some of the connections between Baer subgeometry partitions and several other areas of combinatorial interest, most notably projective sets and flagtransitive translation planes.     

A characterisation of the embeddings of PG(m,q) into PG(n,qr).

Let S_m be an embedding of PG(m,q) into PG(n,q^r),with n < m, such that S_m generates PG(n,q^r). S_m can be obtained as a projection from a (m–n–1)-dimensional subspace V_m–n–1 into a non incident n-dimensional subspace V_n of some strong embedding S of PG(m,q) into PG(m,q^r).     

A new graphic characterization of non singular quadrics of PG(r,q)

LetK be ak-set of class [0, 1,m,n]₁ of anr-dimensional projective Galois space PG(r, q) of orderq. We prove that: Ifr = 2s (s 2),k = _2s–1 and if through each point ofK there are exactlyq ^2(s–1) tangent lines and at most _2s–3 n-secant lines, thenK is a non singular quadric of PG(2s,q). Ifr = 2s–1 (s2),k=_2(s–1) +q ^s–1 and if at each point ofK there are exactlyq ^2s–3 –q ^s–2 tangents and at most _2(s–2)+q ^s–2 n-secant lines, thenK is a hyperbolic quadric of PG(2s–1,q).     

Partitions of Desarguesian and Hughes planes into complete and incomplete arcs

The purpose of the present article is to obtain partitions of Desarguesian projective planes PG (2,q²), and of Hughes planes, too, into a number of q²+q+1 equicardinal arcs. In a previous paper we showed that PG(2,q²) is a disjoint union of arcs as above, which, as it turned out later, were also complete. We now prove that a similar partition can be obtained for the Hughes planes. We also partition Desarguesian and Hughes planes into incomplete arcs with the same number of points.     

A unified construction for c*· c- and L · L*-geometries in projective spaces

We present two new constructions for c^* · c-geometries. The first provides, for each even prime powerq, a flag-transitive c^* · c-geometry of orderq–1 that is embedded in the projective space PG(3,q) and which is related with the Cameron-Fisher extended grids of odd type. The second construction is valid independently of the parity ofq. Forq even, it produces the same geometry as the first construction, and forq odd, two geometries related with some extended grids constructed by Meixner and Pasini.Next, by using some complementary models for c^* and L in a projective plane, we derive from our construction a new family of L · L^*-geometries embedded in PG(3,q). Forq even, these geometries are flag-transitive.     

Special point sets inPG(n,q) and the structure of sets with the maximal number of nuclei

The structure of _{n– 1}-sets inPG(n, q) with more thanq – 1 nuclei is investigated. It is shown that classification of these sets with the maximal numberq ^{n– 1}-q ^{n– 2} of nuclei is equivalent to the classification of (q + l)-sets inPG(2,q) havingq –1 nuclei.Dedicated to Professer Walter Benz for his 60^th birthday     

Evolution in a Gaussian Random Field

We consider an evolution process in a Gaussian random field V(q) with the mean ‹V(q)› = 0 and the correlation function W(|q – q|) ‹V(q)V(q)›, where q ^d and d is the dimension of the Euclidean space ^d . For the value ‹G(q,t;q ₀)›, t > 0, of the Green's function of the evolution equation averaged over all realizations of the random field, we use the Feynman–Kac formula to establish an integral equation that is invariant with respect to a continuous renormalization group. This invariance property allows using the renormalization group method to find an asymptotic expression for ‹G(q,t;q ₀)› as |q – q ₀| and t .     

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.     

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.     

Characterisations of Flock Quadrangles

We characterise the Hermitian and Kantor flock generalized quadrangles of order (q ²,q), q even, (associated with the linear and Fisher–Thas–Walker flocks of a quadratic cone, and the Desarguesian and Betten–Walker translation planes) in terms of a self-dual subquadrangle. Equivalently, we show that a herd which contains a translation oval must be associated with the linear or Fisher–Thas–Walker flock. This result is a consequence of the determination of all functions which satisfy a certain absolute trace equation whose form is remarkably similar to that of an equation arising in recent studies of ovoids in three-dimensional projective space of finite order q.     

The Classification of Projective Planes of Order q3 with Cone-Representations in PG (6, q)

The classification of cone-representations of projective planes of orderq ³ of index 3 and rank 4 (and so in PG(6,^q)) is completed. Any projective plane with a non-spread representation (being a cone-representation of the second kind) is a dual generalised Desarguesian translation plane, as found by Jha and Johnson, and conversely. Indeed, given any collineation of PG(2,^q) with no fixed points, there exists such a projective plane of order q³ , where q is a prime power, that has the second kind of cone-representation of index 3 and rank 4 in PG(6,q). An associated semifield plane of order q ³ is also constructed at most points of the plane. Although Jha and Johnson found this plane before, here we can show directly the geometrical connection between these two kinds of planes.     

Varietà di Segre e ovoidi dello spazio polare Q+(7, q)

Summary In this paper, for q even, we construct an ovoid O ₃ and a spread S of the finite classical polar space Q⁺(7, q) determinated by a hyperbolic quadric Q⁺ of PG(7, q) such that there is a subgroup of PGO ₊ ⁸ (q) isomorphic to PGL₂(q ³), which maps O ₃ in itself and S in S and is 3-transitive on O ₃ and on S; for q>2, S is not a Desarguesian spread of Q⁺(7, q) and O ₃ is a Desarguesian ovoid.

---

Varietà di Segre e ovoidi dello spazio polare Q⁺(7, q)

---

Al Prof. Adriano Barlotti in occasione del suo 60^o compleanno     

Mixed partitions of PG(3,q)

A mixed partition of PG(2n−1,q²) is a partition of the points of PG(2n−1,q²) into (n−1)-spaces and Baer subspaces of dimension 2n−1. In (Bruck and Bose, J. Algebra 1 (1964) 85) it is shown that such a mixed partition of PG(2n−1,q²) can be used to construct a (2n−1)-spread of PG(4n−1,q) and hence a translation plane of order q²ⁿ. In this paper, we provide several new examples of such mixed partitions in the case when n=2.     

A characterization of exterior lines of certain sets of points in PG (2, q)

Let A and B be disjoint sets of points in PG(2, q) the Desarguesian projective plane of order q, with |A|q, |B|=q+1, such that each line through a point of A meets B (just once). Then B is a line.     

Blocking sets and partial spreads in finite projective spaces

A t-blocking set in the finite projective space PG(d, q) with dt+1 is a set of points such that any (d–t)-dimensional subspace is incident with a point of and no t-dimensional subspace is contained in . It is shown that | |q ^t +...+1+q ^t–1q and the examples of minimal cardinality are characterized. Using this result it is possible to prove upper and lower bounds for the cardinality of partial t-spreads in PG(d, q). Finally, examples of blocking sets and maximal partial spreads are given.     

Projective spread sets

In this paper the notion of a spread set for at-spread ofPG(2t+1,q) is generalised and it is shown that certaint-spreads ofPG(n, q) correspond to these generalised spread sets. Then a projective spread set is defined and it is shown that anyt-spread ofPG(n, q) corresponds to a projective spread set. Connections between the spread set and the projective spread set of at-spread are discussed, in particular in the case of at-spread ofPG(2t + 1,q) the spread set and the projective spread set are equivalent, giving a new and straightforward construction of a spread set. The methods developed are used to show, with the aid of a computer, that the 1-packing ofPG(7,2) constructed by Baker is regulus-free.Dedicated to Professor Giuseppe Tallini on the occasion of his 60th birthday     

A characterization of hadamard designs with SL(2,q) acting transitively

Given a hyperoval in a projective plane of even orderq, we can associate a Hadamard 2-design. In the case when is the Desarguesian plane P_2,q ,q=2 ^h ,h>1 and is a regular hyperoval (conic and its nucleus) then a design (q) is obtained. (q) has a point transitive automorphism group isomorphic to PSL(2,q)( SL(2,q)). We classify the designs (q) and P_2h–1,2 (the projective space of dimension 2h–1 overF ₂) among all the designsH with the same parameters as (q) admitting an automorphism groupGSL(2,q) acting transitively the points ofH. We also describe how all such designsH may be constructed and discuss the problem of when two such designs are isomorphic.This research was supported by Science and Engineering Research Council Grant GR/G 03359.     

New Maximal Arcs in Desarguesian Planes

Constructions are described of maximal arcs in Desarguesian projective planes utilizing sets of conics on a common nucleus in PG(2, q). Several new infinite families of maximal arcs in PG(2, q) are presented and a complete enumeration is carried out for Desarguesian planes of order 16, 32, and 64. For each arc we list the order of its stabilizer and the numbers of subarcs it contains. Maximal arcs may be used to construct interesting new partial geometries, 2-weight codes, and resolvable Steiner 2-designs.     

Two-Weight Codes, Partial Geometries and Steiner Systems

Two-weight codes and projectivesets having two intersection sizes with hyperplanes are equivalentobjects and they define strongly regular graphs. We construct projective sets in PG(2m – 1,q) that have the sameintersection numbers with hyperplanes as the hyperbolic quadricQ⁺(2m – 1,q). We investigate these sets; we provethat if q = 2 the corresponding strongly regular graphsare switching equivalent and that they contain subconstituentsthat are point graphs of partial geometries. If m = 4the partial geometries have parameters s = 7, t = 8, = 4 and some of them are embeddable in Steinersystems S(2,8,120).