Some New Maximal Sets of Mutually Orthogonal Latin Squares

Two ways of constructing maximal sets of mutually orthogonal Latin squares are presented. The first construction uses maximal partial spreads in PG(3, 4) \ PG(3, 2) with r lines, where r ∈ {6, 7}, to construct transversal-free translation nets of order 16 and degree r + 3 and hence maximal sets of r + 1 mutually orthogonal Latin squares of order 16. Thus sets of t MAXMOLS(16) are obtained for two previously open cases, namely for t = 7 and t = 8. The second one uses the (non)existence of spreads and ovoids of hyperbolic quadrics Q ⁺ (2m + 1, q), and yields infinite classes of q ^{2n ? 1} ? 1 MAXMOLS(q ²ⁿ ), for n ≥ 2 and q a power of two, and for n = 2 and q a power of three.     

Maximal partial ovoids and maximal partial spreads in hermitian generalized quadrangles

Maximal partial ovoids and maximal partial spreads of the hermitian generalized quadrangles H(3,q²) and H(4,q²) are studied in great detail. We present improved lower bounds on the size of maximal partial ovoids and maximal partial spreads in the hermitian quadrangle H(4,q²). We also construct in H(3,q²), q=2^2h+1, h≥ 1, maximal partial spreads of size smaller than the size q²+1 presently known. As a final result, we present a discrete spectrum result for the deficiencies of maximal partial spreads of H(4,q²) of small positive deficiency δ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 101–116, 2008     

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.     

A remark on symplectic spreads

We first note that each element of a symplectic spread of PG(2n − 1, 2 ^r ) 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, 2 ^r ). As an application, we show that a translation plane of order q ⁿ , 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).     

Subregular Spreads of PG (5, 2e)

The theory of subregular spreads of PG(3,q) was developed by R.H. Bruck (1969, in “Combinatorial Mathematics and Its Applications,” Chap. 27, pp. 426–514. Univ. of North Carolina Press, Chapel Hill). An extension of these results was provided to the higher-dimensional case by the author (1998, Finite Fields Appl.4, 362–380); however, examples of such spreads were only constructed in PG(5, q) for q odd. In this paper, we give a construction of subregular spreads for PG(5, q), where q is even.     

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).     

On the Spectrum of the Sizes of Maximal Partial Line Spreads in <Emphasis Type="Italic">PG</Emphasis>(2<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">n</Emphasis> ≥ 3

A lot of research has been done on the spectrum of the sizes of maximal partial spreads in PG(3,q) [P. Govaerts and L. Storme, Designs Codes and Cryptography, Vol. 28 (2003) pp. 51–63; O. Heden, Discrete Mathematics, Vol. 120 (1993) pp. 75–91; O. Heden, Discrete Mathematics, Vol. 142 (1995) pp. 97–106; O. Heden, Discrete Mathematics, Vol. 243 (2002) pp. 135–150]. In [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129], results on the spectrum of the sizes of maximal partial line spreads in PG(N,q), N 5, are given. In PG(2n,q), n 3, the largest possible size for a partial line spread is q^2n-1+q^2n-3+...+q³+1. The largest size for the maximal partial line spreads constructed in [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129] is (q²ⁿ⁺¹–q)/(q²–1)–q³+q²–2q+2. This shows that there is a non-empty interval of values of k for which it is still not known whether there exists a maximal partial line spread of size k in PG(2n,q). We now show that there indeed exists a maximal partial line spread of size k for every value of k in that interval when q 9.J. Eisfeld: Supported by the FWO Research Network WO.011.96NP. Sziklai: The research of this author was partially supported by OTKA D32817, F030737, F043772, FKFP 0063/2001 and Magyary Zoltan grants. The third author is grateful for the hospitality of Ghent University.     

On the classification of semifield flocks

A classical lemma of Weil is used to characterise quadratic polynomials f with coefficients GF(qⁿ), q odd, with the property that f(x) is a non-zero square for all x∈GF(q). This characterisation is used to prove the main theorem which states that there are no subplanes of order q contained in the set of internal points of a conic in PG(2,qⁿ) for q?4n²−8n+2. As a corollary to this theorem it then follows that the only semifield flocks of the quadratic cone of PG(3,qⁿ) for those q exceeding this bound are the linear flocks and the Kantor-Knuth semifield flocks.     

The packing problem for projective geometries over GF(3) with dimension greater than five

An (n, d) set in the projective geometry PG(r, q) is a set of n points, no d of which are dependent. The packing problem is that of finding n(r, q, d), the largest size of an (n, d) set in PG(r, q). The packing problem for PG(r, 3) is considered. All of the values of n(r, 3, d) for r ? 5 are known. New results for r = 6 are n(6, 3, 5) = 14 and 20 ? n(6, 3, 4) ? 31. In general, upper bounds on n(r, q, d) are determined using a slightly improved sphere-packing bound, the linear programming approach of coding theory, and an orthogonal (n, d) set with the known extremal values of n(r, q, d)—values when r and d are close to each other. The BCH constructions and computer searches are used to give lower bounds. The current situation for the packing problem for PG(r, 3) with r ? 15 is summarized in a final table.     

A trivalent graph with 58 vertices and girth 9

The concept of a partial three-space is due to Laskar and Dunbar, and is a three-dimensional analogue of a partial geometry. Here we determine all partial three-spaces S for which the S-planes are planes of PG(n, q), for which the S-lines are all the lines contained in the S-planes, for which the S-points are all the points in the S-planes, and for which the incidence relation is that of PG(n, q). More generally, we determine all partial three-spaces S for which the S-lines are lines of PG(n,q), for which the S-points are all the points on these lines, and for which the incidence relation is that of PG(n, q).     

Orthogonal packings inPG(5, 2)

Aspread inPG(n, q) is a set of lines which partitions the point set. A packing inPG(n, q) (n odd) is a partition of the lines into spreads. Two packings ofPG(n, q) are calledorthogonal if and only if any two spreads, one from each packing, have at most one line in common. Recently, R. D. Baker has shown the existence of a pair of orthogonal packings inPG(5, 2). In this paper we enumerate all packings inPG(5, 2) having both an automorphism of order 31 and the Frobenius automorphism. We find all pairs of orthogonal packings of the above type and display a set of six mutually orthogonal packings. Previously the largest set of orthogonal packings known inPG(5, 2) was two.     

Blocking sets and semifields

We find a relationship between semifield spreads of PG(3,q), small Rédei minimal blocking sets of PG(2,q²), disjoint from a Baer subline of a Rédei line, and translation ovoids of the hermitian surface H(3,q²).     

Maximal partial spreads in PG (3, q)

We transfer the whole geometry of PG(3, q) over a non-singular quadric Q_4,q of PG(4, q) mapping suitably PG(3, q) over Q_4,q. More precisely the points of PG(3, q) are the lines of Q_4,q; the lines of PG(3, q) are the tangent cones of Q_4,q and the reguli of the hyperbolic quadrics hyperplane section of Q_4,q. A plane of PG(3, q) is the set of lines of Q_4,q meeting a fixed line of Q_4,q. We remark that this representation is valid also for a projective space over any field K and we apply the above representation to construct maximal partial spreads in PG(3, q). For q even we get new cardinalities for For q odd the cardinalities are partially known.     

Almost Desarguesian Maximal Partial Spreads

Constructions are given of various classesof maximal partial spreads in PG(3,2 ^r ) whose partialspreads consist of q/2 reguli sharing a line. Further,characterization results are given for the main classes of constructedmaximal partial spreads.     

Complete Spans on Hermitian Varieties

Let L be a general linear complex in PG(3, q) for any prime power q. We show that when GF(q) is extended to GF(q ²), the extended lines of L cover a non-singular Hermitian surface H ? H(3, q ²) of PG(3, q ²). We prove that if Sis any symplectic spread PG(3, q), then the extended lines of this spread form a complete (q ² + 1)-span of H. Several other examples of complete spans of H for small values of q are also discussed. Finally, we discuss extensions to higher dimensions, showing in particular that a similar construction produces complete (q ³ + 1)-spans of the Hermitian variety H(5, q ²).     

On the Action of the Groups GL(n+1, q), PGL(n + 1, q), SL(n + 1, q) and PSL(n + 1, q) on PG(n, q t)

Let q be a prime power and let n ≥ 0, t ≥ 1 be integers. We determine the sizes of the point orbits of each of the groups GL(n + 1, q), PGL(n + 1, q), SL(n + 1, q) and PSL(n + 1, q) acting on PG(n, q ^t) and for each of these sizes (and groups) we determine the exact number of point orbits of this size.     

Near-MDS codes arising from algebraic curves

Every elliptic quartic Γ₄ of PG(3,q) with nGF(q)-rational points provides a near-MDS code C of length n and dimension 4 such that the collineation group of Γ₄ is isomorphic to the automorphism group of C. In this paper we assume that GF(q) has characteristic p>3. We classify the linear collineation groups of PG(3,q) which can preserve an elliptic quartic of PG(3,q). Also, we prove for q?113 that if the j-invariant of Γ₄ does not disappear, then C cannot be extended in a natural way by adding a point of PG(3,q) to Γ₄.     

On the Twisted Cubic of PG(3, q)

In this paper we classify the lines of PG(3, q) whose points belong to imaginary chords of the twisted cubic of PG(3, q). Relying on this classification result, we obtain a complete classification of semiclassical spreads of the generalized hexagon H(q).     

Characterizing geometric designs, II

We provide a characterization of the classical point-line designs PG₁(n,q), where n?3, among all non-symmetric 2-(v,k,1)-designs as those with the maximal number of hyperplanes. As an application of this result, we characterize the classical quasi-symmetric designs PGn−2(n,q), where n?4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q+1 and all intersection numbers at least qn−4+?+q+1. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as PG₁(n,q); in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.     

Some Characterizations of Finite Hermitian Veroneseans

We characterize the finite Veronesean of all Hermitian varieties of PG(n,q²) as the unique representation of PG(n,q²) in PG(d,q), d n(n+2), where points and lines of PG(n,q²) are represented by points and ovoids of solids, respectively, of PG(d,q), with the only condition that the point set of PG(d,q) corresponding to the point set of PG(n,q²) generates PG(d,q). Using this result for n=2, we show that is characterized by the following properties: (1) ; (2) each hyperplane of PG(8,q) meets in q²+1, q³+1 or q³+q²+1 points; (3) each solid of PG(8,q) having at least q+3 points in common with shares exactly q²+1 points with it.51E24