Unitary Polarity Research Articles

Achromatic colorings of polarity graphs

A complete partition of a graph G is a partition of the vertex set such that there is at least one edge between any two parts. The largest r such that G has a complete partition into r parts, each of which is an independent set, is the achromatic number of G. We determine the achromatic number of polarity graphs of biaffine planes coming from generalized polygons. Our colorings of a family of unitary polarity graphs are used to solve a problem of Axenovich and Martin on complete partitions of C4-free graphs. Furthermore, these colorings prove that there are sequences of graphs which are optimally complete and have unbounded degree, a problem that had been studied for the sequence of hypercubes independently by Roichman, and Ahlswede, Bezrukov, Blokhuis, Metsch, and Moorhouse.

Arbitrary spatially variant polarization by unitary transformations in a common path interferometer

We introduce a novel, common path interferometric technique, for generating hybrid spatially variant polarized fields and Poincaré beams. The technique is based on unitary polarization transformations employing a single spatial light modulator in a Sagnac-like common path interferometer. This technique allows generating arbitrary fields as a superposition of orthogonal elliptically polarized vortex basis with opposite handedness. We demonstrate the generation of such fields theoretically and verify it experimentally for azimuthal and spiral polarization. As an example of application, we generated a radially polarized ring-shaped field and characterize it by Stokes polarimetry.

On the classification problem for the genera of quotients of the Hermitian curve

In this article, we characterize the genera of those quotient curves of the -maximal Hermitian curve for which either G is contained in the maximal subgroup of fixing a self-polar triangle, or q is even and G is contained in the maximal subgroup of fixing a pole-polar pair with respect to the unitary polarity associated to In this way, several new values for the genus of a maximal curve over a finite field are obtained. Our results leave just two open cases to provide the complete list of genera of Galois subcovers of the Hermitian curve; namely, the open cases in [4] when G fixes a point and q is even, and the open cases in [33] when and q is odd.

Blocking semiovals in PG(2,q2), q odd, admitting PGL(2,q) as an automorphism group

A (conic) blocking semioval is a set of points in a projective plane (containing a conic) that is both a blocking set and a semioval. Recently, Dover, Mellinger, and Wantz constructed two new families of conic blocking semiovals in the Desarguesian projective planes of odd order. One of their examples, say S, arises by considering in PG(2,q2), q odd, a unitary polarity commuting with an orthogonal polarity. In particular, such a conic blocking semioval has q3−q2+q2+1 points and is stabilized by a group G isomorphic to PGL(2,q). In this paper, we show that in the same geometric setting there are conic blocking semiovals, having the same size, admitting the same group G, but not isomorphic to S.

Triangle-free induced subgraphs of the unitary polarity graph

Let ⊥ be a unitary polarity of a finite projective plane π of order q2. The unitary polarity graph is the graph with vertex set the points of π where two vertices x and y are adjacent if x∈y⊥. We show that a triangle-free induced subgraph of the unitary polarity graph of an arbitrary projective plane has at most (q4+q)∕2 vertices. When π is the Desarguesian projective plane PG(2,q2) and q is even, we show that the upper bound is asymptotically sharp, by providing an example on q4∕2 vertices. Finally, the case when π is the Figueroa plane is discussed.

On the spectrum of genera of quotients of the Hermitian curve

ABSTRACTWe investigate the genera of quotient curves ℋq∕G of the -maximal Hermitian curve ℋq, where G is contained in the maximal subgroup fixing a pole-polar pair (P,ℓ) with respect to the unitary polarity associated with ℋq. To this aim, a geometric and group-theoretical description of ℳq is given. The genera of some other quotients ℋq∕G with G≰ℳq are also computed. In this way we obtain new values in the spectrum of genera of -maximal curves. The complete list of genera g>1 of quotients of ℋq is given for q≤29, as well as the genera g of quotients of ℋq with for any q. As a direct application, we exhibit examples of -maximal curves which are not Galois covered by ℋq when q is not a cube. Finally, a plane model for ℋq∕G is obtained when G is cyclic of order p⋅d, with d a divisor of q+1.

Embedding of classical polar unitals in PG(2,q2)

A unital, that is, a block-design 2−(q3+1,q+1,1), is embedded in a projective plane Π of order q2 if its points and blocks are points and lines of Π. A unital embedded in PG(2,q2) is Hermitian if its points and blocks are the absolute points and non-absolute lines of a unitary polarity of PG(2,q2). A classical polar unital is a unital isomorphic, as a block-design, to a Hermitian unital. We prove that there exists only one embedding of the classical polar unital in PG(2,q2), namely the Hermitian unital.

Unitals in shift planes of odd order

Abstract A finite shift plane can be equivalently defined via abelian relative difference sets as well as planar functions. In this paper, we present a generic way to construct unitals in finite shift planes of odd square order. We investigate various geometric and combinatorial properties of these planes, such as the self-duality, the existence of O’Nan configurations, Wilbrink’s conditions, the designs formed by circles and so on. We also show that our unitals are inequivalent to the unitals derived from unitary polarities in the same shift planes. As designs, our unitals are also not isomorphic to the classical unitals (the Hermitian curves).

Parabolic unitals in a family of commutative semifield planes

In the commutative semifield planes constructed in Zhou and Pott (2013), we obtain a family of parabolic transitive unitals. For any unital in this family, we prove that there is a collineation group fixing it and acts sharply transitively on its affine points. We also consider its dual unital, the collinearity of its feet and show that as a design it is always resolvable. In particular, we give a necessary and sufficient condition under which a unital in our family is equivalent to a Ganley unital derived from a unitary polarity.

A note on unitary polarities in finite Dickson semifield planes

It is shown that there is only one class of unitary polarities in the Dickson semifield planes and hence one class of polar unitals in such planes.

On the structure of the Figueroa unital and the existence of O’Nan configurations

The finite Figueroa planes are non-Desarguesian projective planes of order q3 for all prime powers q>2, constructed algebraically in 1982 by Figueroa, and Hering and Schaeffer, and synthetically in 1986 by Grundhöfer. All Figueroa planes of finite square order are shown to possess a unitary polarity by de Resmini and Hamilton in 1998, and hence admit unitals. Hui and Wong (2012) have shown that these polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unital to be classical, and hence they are not classical. In this article we introduce and make use of a new alternative synthetic description of the Figueroa plane and unital to demonstrate the existence of O’Nan configurations, thus providing support to Piper’s conjecture (1981).

Hyperoval constructions on the Hermitian surface

New infinite families of hyperovals of the generalized quadrangle H(3,q2) are provided. They arise in different geometric contexts. More precisely, we construct hyperovals by means of certain subsets of the projective plane called here k-tangent arcs with respect to a Hermitian curve (Section 2), hyperovals arising from the geometry of an orthogonal polarity commuting with a unitary polarity (Section 3), hyperovals admitting the irreducible linear group PSL(2,7) as a subgroup of PGU(3,q2), q=ph, p≡3,5or6(mod7) and h an odd integer (Section 4). Finally we construct hyperovals by means of the embedding of PSp(4,q)<PGU(4,q2) as a subfield subgroup (Section 5).

Blind, fast and SOP independent polarization recovery for square dual polarization–MQAM formats and optical coherent receivers

We present both theoretically and experimentally a novel blind and fast method for estimating the State of Polarization (SOP) of a single carrier channel modulated in square Dual Polarization (DP) MQAM format for optical coherent receivers. The method can be used on system startup, for quick channel reconfiguration, or for burst mode receivers. It consists of converting the received waveform from Jones to Stokes space and looping over an algorithm until a unitary polarization derotation matrix is estimated. The matrix is then used to initialize the center taps of the subsequent classical decision-directed stochastic gradient algorithm (DD-LMS). We present experimental comparisons of the initial Bit Error Rate (BER) and the speed of convergence of this blind Stokes space polarization recovery (PR) technique against the common Constant Modulus Algorithm (CMA). We demonstrate that this technique works on any square DP-MQAM format by presenting experimental results for DP-4QAM, -16QAM and -64QAM at varying distances and baud rates. We additionally numerically assess the technique for varying differential group delays (DGD) and sampling offsets on 28 Gbaud DP-4QAM format and show fast polarization recovery for instantaneous DGD as high as 90% of symbol duration. We show that the convergence time of this blind PR technique does not depend on the initial SOP as CMA does and allows switching to DD-LMS faster by more than an order of magnitude. For DP-4QAM, it shows a convergence time of 5.9 ns, which is much smaller than the convergence time of recent techniques using modified CMA algorithms for quicker convergence. BER of the first 20 × 10(3) symbols is always smaller by several factors for DP-16QAM and -64QAM but not always for DP-4QAM.

Non-classical polar unitals in finite Figueroa planes

The finite Figueroa planes are non-Desarguesian projective planes of order q 3 for all prime powers q > 2. These planes were constructed algebraically in 1982 by Figueroa, and Hering and Schaeffer, and synthetically in 1986 by Grundhöfer. All Figueroa planes of finite square order are shown to possess a unitary polarity by de Resmini and Hamilton in 1998, and hence admit unitals. Using the result of O’Nan in 1971 on the non-existence of his configuration in a classical unital, and the intrinsic characterization by Taylor in 1974 of the notion of perpendicularity induced by a unitary polarity in the classical plane (introduced by Dembowski and Hughes in 1965), we show that these Figueroa polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unitary block design to be classical, and hence they are not classical.

Unitals of PG(2,q2) Containing Conics

AbstractA unital in PG(2, q2) is a set of points such that each line meets in 1 or points. The well‐known example is the classical unital consisting of all absolute points of a unitary polarity of PG(2, q2). Unitals other than the classical one also exist in PG(2, q2) for every . Actually, all known unitals are of Buekenhout–Metz type [see F. Buekenhout, Geom Dedicata 5 (1976), 189–194, R. Metz, Geom Dedicata 8 (1979), 125–126.], and they can be obtained by a construction due to F. Buekenhout, (Geom Dedicata 5 (1976), 189–194).. The unitals constructed by R. D. Baker and G. L. Ebert (J Combin Theory Ser A 60 (1992), 67–84), and independently by J. W. P. Hirschfeld and T. Szőnyi (Discrete Math 97 (1991), 229–242), are the union of q conics. Our Theorem 1.1 shows that this geometric property characterizes the Baker–Ebert–Hirschfeld–Szőnyi unitals. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 101–111, 2013

Effect of radial defect lines in the focalization of unitary polarization order light beams

In this letter, we analyze the effect of the defect line of a θ-cell polarization converter on the focalization of a Gaussian laser beam by means of a high numerical aperture microscope objective. This liquid crystal device is frequently used to convert a linearly polarized laser beam into either a radially or azimuthally polarized beam. The line singularity, that defines the cell axis and characterizes these devices, leads to a π-shift on the light polarization in moving from one side of the cell to the other, with respect to the cell axis. The shift, although negligible for light filtering and polarization microscopy, can be crucial in applications where a strong longitudinal component of the focused field is needed, such as in aperturless near-field microscopy. In this work, light distribution simulations as well as experimental investigations of the fields at the focal plane are carried out.

Blocking sets of [formula omitted] with respect to generators

We construct minimal blocking sets with respect to generators on the Hermitian surfaces H ( n , q 2 ) when n and q are both odd. A blocking set arises from q + 1 quadrics in PG ( n , q 2 ) whose polarities commute with a unitary polarity, constructed from the union of Baer sub-quadrics with the common intersections deleted.

Some two-character sets

Some new examples of two-character sets with respect to planes of PG(3, q 2), q odd, are constructed. They arise from three-dimensional hyperbolic quadrics, from the geometry of an orthogonal polarity commuting with a unitary polarity. The last examples arise from the geometry of the unitary group PSU(3, 3) acting on the split Cayley hexagon H(2).

Polarities and unitals in the Coulter–Matthews planes

For a class of finite shift planes introduced by Coulter and Matthews, we give a set of representatives for the isomorphism types, determine all automorphisms and describe all polarities explicitly. The planes in question are the only known examples of finite shift planes that are not translation planes. Each non-desarguesian Coulter---Matthews plane admits precisely two conjugacy classes of orthogonal polarities. In addition, each Coulter---Matthews plane of square order admits exactly one conjugacy class of unitary polarities. We prove that most of the corresponding unitals are not classical.

Some constructions on the Hermitian surface

In the geometric setting of commuting orthogonal and unitary polarities we construct an infinite family of complete (q + 1)2---spans of the Hermitian surface $${\mathcal {H}(3, q^2)}$$ , q odd. A construction of an infinite family of minimal blocking sets of $${\mathcal {H}(3, q^2)}$$ , q odd, admitting PSL 2(q), is also provided.