In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane. The binary linear codes of length 52 spanned by the incidence matrices of 2-$(52,4,1)$ designs associated with previously known and some newly found maximal arcs of degree 4 in projective planes of order 16 are analyzed and classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes. A conjecture concerning the codes of maximal arcs in $PG(2,2^m)$ is formulated.
The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising frommaximal (120,8)-arcs in the known projective planes of order 16 are computed. It is shown that each of these designs is embeddable in a unique way in a projective plane of order 16.
A combinatorial characterization of resolvable Steiner 2-$(v,k,1)$ designs embeddable as maximal arcs in a projective plane of order $(v-k)/(k-1)$ is proved, and a generalization of a conjecture by Andries Brouwer \cite{Br} is formulated.
An $n$ arc in $(k-1)$-dimensional projective space is a set of $n$ points so that no $k$ lie on a hyperplane. In 1988, Glynn gave a formula to count $n$-arcs in the projective plane in terms of simpler combinatorial objects called superfigurations. Several authors have used this formula to count $n$-arcs in the projective plane for $n \le 10$. In this paper, we determine a formula to count $n$-arcs in projective 3-space. We then use this formula to give exact expressions for the number of $n$-arcs in $\mathbb{P}^3(\mathbb{F}_q)$ for $n \le 7$, which are polynomial in $q$ for $n \le 6$ and quas…
This is an expository article detailing results concerning large arcs in finite projective spaces, which attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known old and more recent theorems. The article is mostly self-contained and includes a proof of the most general form of Segre's lemma of tangents and a short proof of the MDS conjecture over prime fields based on this lemma.
In this paper a 2-arc of size 21 in the projective Hjelmslev plane PHG(2,\Z_25) and a 2-arc of size 22 in PHG(2,\F_5[X]/(X^2)) are given. Both arcs are bigger than the 2-arcs previously known in the respective plane. Furthermore, we will give some information on the geometrical structure of the arcs.
An $n$-arc in a projective plane is a collection of $n$ distinct points in the plane, no three of which lie on a line. Formulas counting the number of $n$-arcs in any finite projective plane of order $q$ are known for $n \le 8$. In 1995, Iampolskaia, Skorobogatov, and Sorokin counted $9$-arcs in the projective plane over a finite field of order $q$ and showed that this count is a quasipolynomial function of $q$. We present a formula for the number of $9$-arcs in any projective plane of order $q$, even those that are non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin's formula as…
It is shown that the maximal size of a $2$-arc in the projective Hjelmslev plane over $\mathbb{Z}_{25}$ is $21$, and the $(21,2)$-arc is unique up to isomorphism. Furthermore, all maximal $(20,2)$-arcs in the affine Hjelmslev plane over $\mathbb{Z}_{25}$ are classified up to isomorphism.
It is known that some good linear codes over a finite ring (R-linear codes) arise from interesting point constellations in certain projective geometries. For example, the expurgated Nordstrom-Robinson code, a nonlinear binary [14,6,6]-code which has higher minimum distance than any linear binary [14,6]-code, can be constructed from a maximal 2-arc in the projective Hjelmslev plane over Z_4.
In 1969 Denniston gave a construction of maximal arcs of degree d in Desarguesian projective planes of even order q, for all d dividing q. In 2002 Mathon gave a construction method generalizing the one of Denniston. We will give a new geometric approach to these maximal arcs. This will allow us to count the number of isomorphism classes of Mathon maximal arcs of degree 8 in PG(2,2^h), h prime.
We show that extended cyclic codes over $\mathbb{F}_q$ with parameters $[q+2,3,q]$, $q=2^m$, determine regular hyperovals. We also show that extended cyclic codes with parameters $[qt-q+t,3,qt-q]$, $1<t<q$, determine (cyclic) Denniston maximal arcs. Similarly, cyclic codes with parameters $[q^2+1,4,q^2-q]$ are equivalent to ovoid codes obtained from elliptic quadrics in $PG(3,q)$. Finally, we give new simple presentations of Denniston maximal arcs in $PG(2,q)$ and elliptic quadrics in $PG(3,q)$.
Given any linear code $C$ over a finite field $GF(q)$ we show how $C$ can be described in a transparent and geometrical way by using the associated Bruen-Silverman code. Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes.
It is proved that for every $d\ge 2$ such that $d-1$ divides $q-1$, where $q$ is a power of 2, there exists a Denniston maximal arc $A$ of degree $d$ in $\PG(2,q)$, being invariant under a cyclic linear group that fixes one point of $A$ and acts regularly on the set of the remaining points of ${A}$. Two alternative proofs are given, one geometric proof based on Abatangelo-Larato's characterization of Denniston arcs, and a second coding-theoretical proof based on cyclotomy and the link between maximal arcs and two-weight codes.
In 1974, J. Thas constructed a new class of maximal arcs for the Desarguesian plane of order $q^2$. The construction relied upon the existence of a regular spread of tangent lines to an ovoid in $\PG(3,q)$ and, in particular, it does apply to the Suzuki--Tits ovoid. In this paper, we describe an algorithm for obtaining a possible representation of such arcs in $\PG(2,q^2)$.
A \textit{k}-arc in the projective space ${\rm PG}(n,q)$ is a set of $k$ projective points such that no subcollection of $n+1$ points is contained in a hyperplane. In this paper, we construct new $60$-arcs and $110$-arcs in ${\rm PG}(4,q)$ that do not arise from rational or elliptic curves. We introduce computational methods that, when given a set $\mathcal{P}$ of projective points in the projective space of dimension $n$ over an algebraic number field $\mathcal{Q}(ξ)$, determines a complete list of primes $p$ for which the reduction modulo $p$ of $\mathcal{P}$ to the projective space ${\rm PG…
N. Hamilton and J. A. Thas describe a link between maximal arcs of Mathon type and partial flocks of the quadratic cone. This link is of a rather algebraic nature. In this paper we establish a geometric connection between these two structures. We also define a composition on the flock planes and use this to work out an analogue of the synthetic version of Mathon's Theorem. Finally, we show how it is possible to construct a maximal arc of Mathon type of degree 2d, containing a Denniston arc of degree d provided that there is a solution to a certain given system of trace conditions.
In a recent paper [M], Mathon gives a new construction of maximal arcs which generalizes the construction of Denniston. In relation to this construction, Mathon asks the question of determining the largest degree of a non-Denniston maximal arc arising from his new construction. In this paper, we give a nearly complete answer to this problem. Specifically, we prove that when $m\geq 5$ and $m\neq 9$, the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,1}-map is $(\floor {m/2} +1)$. This confirms our conjecture in [FLX]. For {p,q}-maps, we prove that if $…
In a former paper the authors counted the number of non-isomorphic Mathon maximal arcs of degree 8 in PG(2,2^h), h not 7 and prime. In this paper we will show that in PG(2,2^7) a special class of Mathon maximal arcs of degree 8 arises which admits a Singer group (i.e. a sharply transitive group) on the 7 conics of these arcs. We will give a detailed description of these arcs, and then count the total number of non-isomorphic Mathon maximal arcs of degree 8. Finally we show that the special arcs found in PG(2,2^7) extend to two infinite families of Mathon arcs of degree 8 in PG(2,2^k), k odd an…
The projective degrees of strict partitions of n were computed for all n < 101 and the partitions with maximal projective degree were found for each n. It was observed that maximizing partitions for successive values of n "lie close to each other" in a certain sense. Conjecturing that this holds for larger values of n, the partitions of maximal degree were computed for all n < 221. The results are consistent with a recent conjecture on the limiting shape of the strict partition of maximal projective degree.
In this paper we construct functional codes from Denniston maximal arcs. For $q=2^{4n+2}$ we obtain linear codes with parameters $[(\sqrt{q}-1)(q+1),5,d]_q$ where $\lim_{q \to +\infty} d=(\sqrt{q}-1)q-3\sqrt{q}$. We also find for $q=16,32$ a number of linear codes which appear to have larger minimum distance with respect to the known codes with same length and dimension.
A lower bound on the minimum degree of the plane algebraic curves containing every point in a large point-set $K$ of the Desarguesian plane $PG(2,q)$ is obtained. The case where $K$ is a maximal $(k,n)$-arc is considered to greater extent.
We introduce and study embeddings of graphs in finite projective planes, and present related results for some families of graphs including complete graphs and complete bipartite graphs. We also make connections between embeddings of graphs and the existence of certain substructures in a plane, such as Baer subplanes and arcs.
The subject of this paper are partial geometries $pg(s,t,α)$ with parameters $s=d(d'-1), \ t=d'(d-1), \ α=(d-1)(d'-1)$, $d, d' \ge 2$. In all known examples, $q=dd'$ is a power of 2 and the partial geometry arises from a maximal arc of degree $d$ or $d'$ in a projective plane of order $q$ via a known construction due to Thas \cite{Thas73} and Wallis \cite{W}, with a single known exception of a partial geometry $pg(4,6,3)$ found by Mathon \cite{Math} that is not associated with a maximal arc in the projective plane of order 8. A parallel class of lines is a set of pairwise disjoint lines that c…
An (n,r)-arc in PG(2,q) is a set of n points such that each line contains at most r of the selected points. It is well-known that (n,r)-arcs in PG(2,q) correspond to projective linear codes. Let m_r(2,q) denote the maximal number n of points for which an (n,r)-arc in PG(2,q) exists. In this paper we obtain improved lower bounds on m_r(2,q) by explicitly constructing (n,r)-arcs. Some of the constructed (n,r)-arcs correspond to linear codes meeting the Griesmer bound. All results are obtained by integer linear programming.
We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal non-percolating sets. Additional results on the minimal and maximal percolation time as well as on the critical probability in the projective plane are also presented.
Bishnoi conjectured that if a minimal t-fold blocking set in a projective plane of prime power order has maximal size then it is either a projective plane minus one point, the complement of a Baer subplane or a unital. In this note we prove this conjecture.
For maximal planar graphs of order $n\geq 4$, we prove that a vertex--coloring containing no rainbow faces uses at most $\lfloor\frac{2n-1}{3}\rfloor$ colors, and this is best possible. For maximal graph embedded on the projective plane, we obtain the analogous best bound $\lfloor\frac{2n+1}{3}\rfloor$. The main ingredients in the proofs are classical homological tools. By considering graphs as topological spaces, we introduce the notion of a null coloring, and prove that for any graph $G$ a maximal null coloring $f$ is such that the quotient graph $G/f$ is a forest.
In this article, we analyse maximal sets of $k$-spaces, in PG(n,q) and AG(n,q), $n>2k+t+2$, that pairwise meet in at least a $t$-space. It is known that for both PG(n,q) and AG(n,q), the largest example is a $t$-pencil, i.e. the set of all $k$-spaces containing a fixed $t$-space. In this paper, we analyse the structure of the second largest maximal example in both PG(n,q) and AG(n,q).
We show that the Ree unital $\mathcal{R}(q)$ has an embedding in a projective plane over a field $F$ if and only if $q=3$ and $\mathbb{F}_8$ is a subfield of $F$. In this case, the embedding is unique up to projective linear transformations. Besides elementary calculations, our proof uses the classification of the maximal subgroups of the simple Ree groups.
In the previous works of the authors, a step-by-step algorithm FOP which uses any fixed order of points in the projective plane $\mathrm{PG}(2,q)$ is proposed to construct small complete arcs. In each step, the algorithm adds to a current arc the first point in the fixed order not lying on the bisecants of the arc. The algorithm is based on the intuitive postulate that $\mathrm{PG}(2,q)$ contains a sufficient number of relatively small complete arcs. Also, in the previous papers, it is shown that the type of order on the points of $\mathrm{PG}(2,q)$ is not relevant. A complete lexiarc in $\mat…
Given a marked surface (S,M) we can add arcs to the surface to create a triangulation, T, of that surface. For each triangulation, T, we can associate a cluster algebra. In this paper we will consider the torus of genus n with two interior marked points (called punctures). We will construct a specific triangulation of this surface which yeilds a specific quiver. Then in the sense of work by Keller we will produce a maximal green sequence for this quiver.
We characterise mutations between fake weighted projective spaces, and give explicit formulas for how the weights and multiplicity change under mutation. In particular, we prove that multiplicity-preserving mutations between fake weighted projective spaces are mutations over edges of the corresponding simplices. As an application, we analyse the canonical and terminal fake weighted projective spaces of maximal degree.
In this paper we prove an Erdős-Ko-Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group PGL(3,q), in its natural action on the points of the projective line, is either a coset of the stabilizer of a point or a coset of the stabilizer of a line.
