Algorithms for Finding Unitals and Maximal Arcs in Projective Planes of Order 16

The paper has been presented at the International Conference Pioneers of Bulgarian Mathematics, Dedicated to Nikola Obreshkoff and Lubomir Tschakalo ff , Sofia, July, 2006.Two heuristic algorithms (M65 and M52) for finding respectively unitals and maximal arcs in projective planes of order 16 are described. The exact algorithms based on exhaustive search are impractical because of the combinatorial explosion (huge number of combinations to be checked). Algorithms M65 and M52 use unions of orbits of di erent subgroups of the automorphism group of the 273x273 bipartite graph of the projective plane. Two very efficient algorithms (developed by the author and not described here) are used in M65 and M52: (i) algorithm VSEPARN for computing the generators, orbits and order of the graph automorphism group; (ii) graph isomorphism algorithm derived from VSEPARN. Four properties are proved and used to speed up the algorithms M65 and M52. The results of these algorithms are published. After changing only the parameters of these algorithms they can be used for determining unitals in projective planes of different orders

Collineation groups of translation planes of small dimension

A subgroup of the linear translation complement of a translation plane is geometrically irreducible if it has no invariant lines or subplanes. A similar definition can be given for geometrically primitive. If a group is geometrically primitive and solvable then it is fixed point free or metacyclic or has a normal subgroup of order w2a+b where wa divides the dimension of the vector space. Similar conditions hold for solvable normal subgroups of geometrically primitive nonsolvable groups. When the dimension of the vector space is small there are restrictions on the group which might possibly be in the translation complement. We look at the situation for certain orders of the plane

On Translation Hyperovals in Semifield Planes

In this paper we demonstrate the first example of a finite translation plane which does not contain a translation hyperoval, disproving a conjecture of Cherowitzo. The counterexample is a semifield plane, specifically a Generalised Twisted Field plane, of order $64$. We also relate this non-existence to the covering radius of two associated rank-metric codes, and the non-existence of scattered subspaces of maximum dimension with respect to the associated spread

Abstract hyperovals, partial geometries, and transitive hyperovals

Includes bibliographical references.2015 Summer.A hyperoval is a (q+2)- arc of a projective plane π, of order q with q even. Let G denote the collineation group of π containing a hyperoval Ω. We say that Ω is transitive if for any pair of points x, y is an element of Ω, there exists a g is an element of G fixing Ω setwise such that xg = y. In1987, Billotti and Korchmaros proved that if 4||G|, then either Ω is the regular hyperoval in PG(2,q) for q=2 or 4 or q = 16 and |G||144. In 2005, Sonnino proved that if |G| = 144, then π is desarguesian and Ω is isomorphic to the Lunelli-Sce hyperoval. For our main result, we show that if G is the collineation group of a projective plane containing a transitivehyperoval with 4 ||G|, then |G| = 144 and Ω is isomorphic to the Lunelli-Sce hyperoval. We also show that if A(X) is an abstract hyperoval of order n ≡ 2(mod 4); then |Aut(A(X))| is odd. If A(X) is an abstract hyperoval of order n such that Aut(A(X)) contains two distinct involutions with |FixX(g)| and |FixX(ƒ)| ≥ 4. Then we show that FixX(g) ≠ FixX(ƒ). We also show that there is no hyperoval of order 12 admitting a group whose order is divisible by 11 or 13, by showing that there is no partial geometry pg(6, 10, 5) admitting a group of order 11 or of order 13. Finally, we were able to show that there is no hyperoval in a projective plane of order 12 with a dihedral subgroup of order 14, by showing that that there is no partial geometry pg(7, 12, 6) admitting a dihedral group of order 14. The latter results are achieved by studying abstract hyperovals and their symmetries

A survey on Traitor Tracing Schemes

When intellectual properties are distributed over a broadcast network, the content is usually encrypted in a way such that only authorized users who have a certain set of keys, can decrypt the content. Some authorized users may be willing to disclose their keys in constructing a pirate decoder which allows illegitimate users to access the content. It is desirable to determine the source of the keys in a pirate decoder, once one is captured. Traitor tracing schemes were introduced to help solve this problem. A traitor tracing scheme usually consists of: a scheme to generate and distribute each user's personal key, a cryptosystem used to protect session keys that are used to encrypt/decrypt the actual content, and a tracing algorithm to determine one source of the keys in a pirate decoder. In this thesis, we survey the traitor tracing schemes that have been suggested. We group the schemes into two groups: symmetric in which the session key is encrypted and decrypted using the same key and asymmetric schemes in which the session key is encrypted and decrypted using different keys. We also explore the possibility of a truly public scheme in which the data supplier knows the encryption keys only. A uniform analysisis presented on the efficiency of these schemes using a set of performance parameters

A Note on Line-Baer subspace Partitions of PG(3, 4)

We consider the partitioning of PG(3, 4) into two types of objects, lines and Baer sub-3-spaces. Any such mixed partition gives rise to a spread of PG(7, 2) (and hence a projective plane of order 16) via a construction tech-nique given in [6]. The author has used the software package Magma to de-termine all such mixed partitions up to equivalence. It turns out that all of the translation planes of order 16 arise from one of the mixed partitions. Mathematics Subject Classification (2000): 51E2

Maximal arcs in projective planes of order 16 and related designs

The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52,4,1) designs arising from previously known maximal (52,4)-arcs, as well as some newly discovered maximal (52,4)-arcs in the known projective planes of order 16, are computed. It is shown that each 2-(120,8,1) design associated with a maximal (120,8)-arc is embeddable in a unique way in a projective plane of order 16. This result suggests a possible strengthening of the Bose–Shrikhande theorem about the embeddability of the complement of a hyperoval in a projective plane of even order. The computations of the maximal sets of compatible resolutions of the 2-(52,4,1) designs associated with maximal (52,4)-arcs show that five of the known projective planes of order 16 contain maximal arcs whose associated designs are embeddable in two nonisomorphic planes of order 16