Hermitian Unitals Research Articles

An isomorphism between unitals and between related classical groups

Abstract An isomorphism between two hermitian unitals is provided, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).

Finite subunitals of the Hermitian unitals

Every finite subunital of any generalized hermitian unital is itself a hermitian unital; the embedding is given by an embedding of quadratic field extensions. In particular, a generalized hermitian unital with a finite subunital is a hermitian one (i.e., it originates from a separable field extension).

Embeddings of hermitian unitals into pappian projective planes

Every embedding of a hermitian unital with at least four points on a block into any pappian projective plane is standard, i.e. it originates from an inclusion of the pertinent fields. This result about embeddings also allows us to determine the full automorphism groups of (generalized) hermitian unitals.

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.

A non-classical unital of order four with many translations

We give a general construction for unitals of order q admitting an action of SU(2,q). The construction covers the classical Hermitian unitals, Grüning’s unitals in Hall planes and at least one unital of order four where the translation centers fill precisely one block. For the latter unital, we determine the full group of automorphisms and show that there are no group-preserving embeddings into (dual) translation planes of order 16.

Unitary Graphs

Unitary graphs are arc-transitive graphs with vertices the flags of Hermitian unitals and edges defined by certain elements of the underlying finite fields. They played a significant role in a recent classification of a class of arc-transitive graphs that admit an automorphism group acting imprimitively on the vertices. In this paper we prove that all unitary graphs are connected of diameter two and girth three. Based on this we obtain, for any prime power $q > 2$, a lower bound of order $O(\Delta^{5/3})$ on the maximum number of vertices in an arc-transitive graph of degree $\Delta = q(q^2-1)$ and diameter two.

Unitals Admitting All Translations

AbstractAmong all 2‐‐designs, we characterize the Hermitian unitals by the existence of sufficiently many translations. In arbitrary 2‐‐designs, each group of translations with given center acts semiregularly on the set of points different from the center.

Veronesean embeddings of Hermitian unitals

In this paper, we determine the Veronesean embeddings of Hermitian unitals, i.e., the representations of Hermitian unitals as points of a 7-dimensional projective space where the blocks are plane ovals. As an application, we derive that the following objects coincide: (1) the generic hyperplane sections of Hermitian Veroneseans in an 8-dimensional projective space, (2) the Grassmannians of the classical spreads of non-degenerate quadrics of Witt index 2 in a 5-dimensional projective space, (3) the sets of absolute points of trialities of Witt index 1. As a consequence, we prove that the set of absolute points of a triality without fixed lines, but with absolute points, determines the triality quadric and the triality itself uniquely.

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).

Ruled cubic surfaces in PG(4, q), Baer subplanes of PG(2, q2) and Hermitian curves

In PG(2, q 2) let ℓ ∞ denote a fixed line, then the Baer subplanes which intersect ℓ ∞ in q+1 points are called affine Baer subplanes. Call a Baer subplane of PG(2, q 2) non-affine if it intersects ℓ ∞ in a unique point. It is shown by Vincenti (Boll. Un. Mat. Ital. Suppl. 2 (1980) 31) and Bose et al. (Utilitas Math. 17 (1980) 65) that non-affine Baer subplanes of PG(2, q 2) are represented by certain ruled cubic surfaces in the André/Bruck and Bose representation of PG(2, q 2) in PG(4, q) (Math. Z. 60 (1954) 156; J. Algebra 1 (1964) 85; J. Algebra 4 (1966) 117). The André/Bruck and Bose representation of PG(2, q 2) involves a regular spread in PG(3, q). For a fixed regular spread S , it is known that not all ruled cubic surfaces in PG(4, q) correspond to non-affine Baer subplanes of PG(2, q 2) in this manner. In this paper, we prove a characterisation of ruled cubic surfaces in PG(4, q) which represent non-affine Baer subplanes of the Desarguesian plane PG(2, q 2). The characterisation relies on the ruled cubic surfaces satisfying a certain geometric condition. This result and the corollaries obtained are then applied to give a geometric proof of the result of Metsch (London Mathematical Society Lecture Note Series, Vol. 245, Cambridge University Press, Cambridge, 1997, p. 77) regarding Hermitian unitals; a result which was originally proved in a coordinate setting.

Some Applications of Magma in Designs and Codes: Oval Designs, Hermitian Unitals and Generalized Reed–Muller Codes

We describe three applications of Magma to problems in the area of designs and the associated codes: • Steiner systems, Hadamard designs and symmetric designs arising from an oval in an even-order plane, leading in the classical case to bent functions and difference-set designs; • the Hermitian unital as a 2-(q3+ 1, q+ 1, 1) design, and the code overFp where p divides q+ 1; • a basis of minimum-weight vectors for the code over Fpof the design of points and hyperplanes of the affine geometry AGd(Fp), where p is a prime.

Characterizing the Hermitian and Ree Unitals on 28 Points

We prove a conjecture of Brouwer, namely that a 2-(28,4,1) design has 2–rank at least 19, with equality occuring if and only if the design is the Ree unital. We give a similar characterization of the Hermitian unital.

A (c. L*-Geometry for the Sporadic Group J2

We prove the existence of a rank three geometry admitting the Hall–Janko group J2 as flag-transitive automorphism group and Aut(J2) as full automorphism group. This geometry belongs to the diagram (c·L*) and its nontrivial residues are complete graphs of size 10 and dual Hermitian unitals of order 3.

Small blocking sets of hermitian designs

A Hermitian design H(q) consists of the points and Hermitian unitals of PG(2, q 2). A committee of H(q) is a blocking set of H(q) of minimum cardinality b(H(q)). It is proved that the committees of H(3) are the lines of PG(2, 9) and, for all odd q, that 2 q + 2 ≤ b( H( q)) < (1 + 7 ln q)( q 2 + 1) q −1.

On symmetric and quasi-symmetric designs with the symmetric difference property and their codes

The four symmetric 2-(64, 28, 12) designs with the symmetric difference property are characterized as the only designs with the given parameters and minimal rank over GF(2). These designs give non-isomorphic quasi-symmetric 2-(36, 16, 12) and 2-(28, 12, 11) designs as residual and derived designs. The binary codes of the quasi-symmetric 2-(28, 12, 11) designs provide four inequivalent self-orthogonal doubly-even (28, 7, 12) codes. This gives a negative answer to the question for the uniqueness of the code of the Hermitian unital of order 3.

Hermitian unitals are code words

We show that a unital PG(2, q 2) is Hermitian if and only if it is in the code generated by the lines of PG(2, q 2). This implies the truth of a conjecture made by Assmus and Key.

Unitals in the H�lz design on 28 points

We show that any unital in the 2-(28, 4, 5) Holz design related to the Hermitian unital of order 3, is of either Hermitian or Ree type.

Arcs and Ovals in the Hermitian and Ree Unitals

The hermitian unitals U ( q ) and the Ree unitals RU ( q ) are examined for the existence of ovals and arcs. It is shown that U ( q ) does not have ovals for q > 2 and that RU ( q ), like U ( q ), is embedded in a much larger design with block intersections of cardinality ⩽ 2. Arcs of size 3 q + 1 are constructed for the Ree unitals RU ( q ); they are ovals only in the case q = 3. In this case, U (3) and RU (3) are embedded in the same design and its automorphism group, the symplectic group Sp (6, 2), contains the automorphism groups of both the unitals; the coding-theoretic aspects are elucidated.