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 Grundh?fer. 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.     

An upper bound for the minimum weight of dual codes of Figueroa planes

In this note we give an explicit construction for words of weight 2q³ - q² - q in the dual p-ary code of the Figueroa plane of order q³, where q > 2 is any power of the prime p. When p is odd this then allows us, for the Figueroa planes, to improve on the previously known upper bound of 2q³ for the minimum weight of the dual p-ary code of any plane of order q³. The construction is the same as one that applies to desarguesian planes of order q³ as described in [3].     

The existence of maximal (q 2, 2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic

We prove that (q ², 2)-arcs exist in the projective Hjelmslev plane PHG(2, R) over a chain ring R of length 2, order |R| = q ² and prime characteristic. For odd prime characteristic, our construction solves the maximal arc problem. For characteristic 2, an extension of the above construction yields the lower bound q ² + 2 on the maximum size of a 2-arc in PHG(2, R). Translating the arcs into codes, we get linear [q ³, 6, q ³ ?q ² ?q] codes over ${\mathbb {F}_q}$ for every prime power q > 1 and linear [q ³ + q, 6,q ³ ?q ² ?1] codes over ${\mathbb {F}_q}$ for the special case q = 2 ^r . Furthermore, we construct 2-arcs of size (q + 1)²/4 in the planes PHG(2, R) over Galois rings R of length 2 and odd characteristic p ².     

A class of non-Desargusian planes

A new class of non-Desargusian planes of order q², where q is a power of an odd prime, is constructed. These planes have the interesting property that they all admit a collineation group of order (q² ? 1).     

On small {;k; q};-arcs in planes of order q2

In this paper we examine some properties of complete {;k; q};-arcs in projective planes of order q². In particular, we derive a lower bound for k, and we exhibit a family of arcs having low values of k which exist in every such plane having a Baer subplane. In addition we resolve the existence problem for complete {;k; 3};-arcs in PG(2, 9).     

Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q ³), q = p ^h , p ≥ 7 prime, of size less than 3(q ^3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size less than 3(p ^3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q ³) which meet every (n ? 2)-space in 1 modulo q points.     

Small point sets of PG(n, p 3h ) intersecting each line in 1 mod p h points

The main result of this paper is that point sets of PG(n, q), q = p ^3h , p ≥ 7 prime, of size < 3(q ^n-1 + 1)/2 intersecting each line in 1 modulo ${\sqrt[3] q}$ points (these are always small minimal blocking sets with respect to lines) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size < 3(p ^3(n-1) + 1)/2 with respect to lines are always linear.     

Translation planes constructed by multiple hyper-regulus replacement

New classes of mutually disjoint hyper-reguli of order q ⁿ , for n > 2, are determined, which are not André hyper-reguli if n > 3. If n is odd, each hyper-regulus permits at least two replacements and if q is odd and (n, q?1) = 1, there are (q?1)/2 mutually disjoint hyper-reguli each of which may be replaced at least two ways. Each of the possible 2^(q-1)/2 translation planes constructed is not André or generalized André. There are also new constructions of mixed subgeometry partitions.     

On a problem of Neumann

A conjecture widely attributed to Neumann is that all finite non-desarguesian projective planes contain a Fano subplane. In this note, we show that any finite projective plane of even order which admits an orthogonal polarity contains many Fano subplanes. The number of planes of order less than $n$ previously known to contain a Fano subplane was $O(logn)$, whereas the number of planes of order less than $n$ that our theorem applies to is not bounded above by any polynomial in $n$.     

Translation planes of order q 2 admitting a two-transitive orbit of length q + 1 on the line at infinity

A classification is given of all translation planes of order q ² that admit a collineation group G admitting a two-transitive orbit of q + 1 points on the line at infinity.     

A class of unitals of order q which can be embedded in two different planes of order q2

By deriving the desarguesian plane of order q² for every prime power q a unital of order q is constructed which can be embedded in both the Hall plane and the dual of the Hall plane of order q² which are non-isomorphic projective planes. The representation of translation planes in the fourdimensional projective space of J. André and F. Buekenhouts construction of unitals in these planes are used. It is shown that the full automorphism groups of these unitals are just the collineation groups inherited from the classical unitals.     

Perfect Baer Subplane Partitions and Three-Dimensional Flag-Transitive Planes

The classification of perfectBaer subplane partitions of PG(2, q²) is equivalentto the classification of 3-dimensional flag-transitive planeswhose translation complements contain a linear cyclic group actingregularly on the line at infinity. Since all known flag-transitiveplanes admit a translation complement containing a linear cyclicsubgroup which either acts regularly on the points of the lineat infinity or has two orbits of equal size on these points,such a classification would be a significant step towards theclassification of all 3-dimensional flag-transitive planes. Usinglinearized polynomials, a parametric enumeration of all perfectBaer subplane partitions for odd q is described.Moreover, a cyclotomic conjecture is given, verified by computerfor odd prime powers q < 200, whose truth would implythat all perfect Baer subplane partitions arise from a constructionof Kantor and hence the corresponding flag-transitive planesare all known.     

Two-dimensional flag-transitive planes revisited

This paper shows that the odd order two-dimensional flag-transitive planes constructed by Kantor-Suetake constitute the same family of planes as those constructed by Baker-Ebert. Moreover, for orders satisfying a modest number theoretical assumption this family consists of all possible such planes of that order. In particular, it is shown that the number of isomorphism classes of (non-Desarguesian) two-dimensional flag-transitive affine planes of order q ² is precisely (q–1)/2 when q is an odd prime and precisely (q–1)/2e when q=p ^e is an odd prime power with exponent e that is a power of 2. An enumeration is given in other cases that uses the Möbius inversion formula.This work was partially supported by NSA grant MDA 904-95-H-1013.This work was partially supported by NSA grant MDA 904-94-H-2033.     

A New Characterization of PSL(p,q)

Abstract

Order components of a finite group are introduced in Chen [Chen, G. Y. (1996c) On Thompson's conjecture. J. Algebra 185:184–193]. It was proved that PSL(3, q), where q is an odd prime power, is uniquely determined by its order components [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002a). A characterization of PSL(3, q) where q is an odd prime power. J. Pure Appl. Algebra 170(2–3): 243–254]. Also in Iranmanesh et al. [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002b). A characterization of PSL(3, q) where q = 2 ⁿ . Acta Math. Sinica, English Ser. 18(3):463–472] and [Iranmanesh, A., Alavi, S. H. (2002). A characterization of simple groups PSL(5, q). Bull. Austral. Math. Soc. 65:211–222] it was proved that PSL(3, q) for q = 2 ⁿ and PSL(5, q) are uniquely determined by their order components. In this paper we prove that PSL(p, q) can be uniquely determined by its order components, where p is an odd prime number. A main consequence of our results is the validity of Thompson's conjecture for the groups under consideration.     

Regular Hjelmslev planes

A projective Hjelmslev plane is called regular iff it admits an Abelian collineation group that is regular on both the points and lines of the plane and that splits into a summand regular on the elements of any given neighborhood and another summand permuting the points and lines of the projective image plane regularly. Regular Hjelmslev planes are shown to correspond to so-called special difference sets. We construct regular Hjelmslev planes with parameters (qⁿ, q) for any prime power q and any natural number n as well as for infinitely many series of parameters (t, q), where t is not a power of q. Our construction also yields series of parameters for which the existence of a Hjelmslev plane was not known up to now as well as the first information on the existence of nontrivial collineations in the case of parameters (t, q) with t not a power of q.     

On translation planes of orderq 3 that admit a collineation group of orderq 3

Let II be a translation plane of orderq ³, with kernel GF(q) forq a prime power, that admits a collineation groupG of orderq ³ in the linear translation complement. Moreover, assume thatG fixes a point at infinity, acts transitively on the remaining points at infinity andG/E is an abelian group of orderq ², whereE is the elation group ofG.In this article, we determined all such translation planes. They are (i) elusive planes of type I or II or (ii) desirable planes.Furthermore, we completely determined the translation planes of orderp ³, forp a prime, admitting a collineation groupG of orderp ³ in the translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity. They are (i) semifield planes of orderp ³ or (ii) the Sherk plane of order 27.     

A note on rational normal curves totally tangent to a Hermitian variety

Let q be a power of a prime integer p, and let X be a Hermitian variety of degree q + 1 in the n-dimensional projective space. We count the number of rational normal curves that are tangent to X at distinct q + 1 points with intersection multiplicity n. This generalizes a result of Segre on the permutable pairs of a Hermitian curve and a smooth conic.     

A characterization of the desarguesian and the Figueroa planes of order $$q^{3}$$

Let \(\Pi \) be a plane of order \(q^{3}\), \(q>2\), admitting \(G\cong PGL(3,q)\) as a collineation group. By Dempwolff (Geometriae Dedicata 18:101–112, 1985) the plane \(\Pi \) contains a G-invariant subplane \(\pi _{0}\) isomorphic to PG(2, q) on which G acts 2-transitively. In this paper it is shown that, if the homologies of \(\pi _{0}\) contained in G extend to \(\Pi \) then \(\Pi \) is either the desarguesian or the Figueroa plane.     

On large maximal partial ovoids of the parabolic quadric Q(4, q)

We use the representation ${T_2(\mathcal{O})}$ for Q(4, q) to show that maximal partial ovoids of Q(4, q) of size q ² ? 1, q = p ^h , p an odd prime, h > 1, do not exist. Although this was known before, we give a slightly alternative proof, also resulting in more combinatorial information of the known examples for q an odd prime.     

Minimal quadratic residue cyclic codes of length 2 n

LetF be a finite field of prime power orderq(odd) and the multiplicative order ofq modulo 2 ⁿ (n>1) be ?(2 ⁿ )/2. Ifn>3, thenq is odd number(prime or prime power) of the form 8m±3. Ifq=8m?3, then the ring $$R_{2^n } = F\left[ x \right]/< x^{2^n } - 1 > $$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length 2 ⁿ generated by these idempotents are completely described. Ifq=8m+3 then the expressions for the 2n?1 primitive idempotents ofR ₂ ⁿ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n?1 idempotents are also obtained. The casen=2, 3 is dealt separately.