A. Aguglia, L. Giuzzi
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 $\PG(2,q^2)$.
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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.
Maximal arcs in small projective Hjelmslev geometries are classified up to isomorphism, and the parameters of the associated codes are determined.
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.
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.
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 $PG(3,q)$.
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.
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…
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.
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…
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.
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…
Constructing maximal pipedreams of double Grothendieck polynomials
本文针对 Pechenik 等人提出的寻找与 rajcode 统计量对应的唯一管道图直接构造方法问题，给出了一种基于阶梯移动的构造算法。
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.
An $(n,r)$-arc in $PG(2,q)$ is a set $B$ of points in $PG(2,q)$ such that each line in $PG(2,q)$ contains at most $r$ elements of $B$ and such that there is at least one line containing exactly $r$ elements of $B$. The value $m_r(2,q)$ denotes the maximal number $n$ of points in the projective geometry $PG(2,q)$ for which an $(n,r)$-arc exists. By explicitly constructing $(n,r)$-arcs using prescribed automorphisms and integer linear programming we obtain some improved lower bounds for $m_r(2,q)$: $m_{10}(2,16)\ge 144$, $m_3(2,25)\ge 39$, $m_{18}(2,25)\ge 418$, $m_9(2,27)\ge 201$, $m_{14}(2,29)…
When Arcs Extend Uniquely: A Higher-Dimensional Generalization of Barlotti's Result
本研究将Barlotti的射影平面弧唯一延拓经典结果推广到高维射影空间，给出高维弧唯一延拓为极大弧的条件，关联A^sMDS码理论，确立线性码唯一延拓为极大长度射影码的条件。
Constructing Maximal Bumpless Pipedreams for Double Grothendieck Polynomials
本文针对双Grothendieck多项式，提出一种显式算法来构造具有给定Rajchgot码行权和列权的最大无碰撞管道流。
Blocking Amalgamations, Maximal Arcs, and Generalized Crowns
该研究针对$r$一致超图，定义$\rho_{r,k}$并证明其上下界，结合截断射影平面、极大$h$-弧等构造确定其精确值，为超图极值理论提供关键结果。
On Projective Planes of Order 16 Associated with 1-rotational 2-(52, 4, 1) Designs
本研究借助MAGMA软件分析22个非同构1-旋转2-(52,4,1)设计在16阶射影平面4次极大弧中的可嵌入性，发现仅1个可嵌入Desarguesian平面PG(2,16)。
A 2-tree is a graph that can be formed by starting with a triangle and iterating the operation of making a new vertex adjacent to two adjacent vertices of the existing graph. Leizhen Cai asked in 1995 whether every maximal planar graph contains a spanning 2-tree. We answer this question in the negative by constructing an infinite class of maximal planar graphs that have no spanning 2-tree.
In the present paper we consider the problem of constructing all the projective rooted spanning trees of a given graph. We propose an algorithm based on reducing this problem to the problem of constructing all the maximal independent sets of a certain derived graph. We also offer an algorithm for step-by-step growing the spanning trees that sifts, at each step, any sub-trees non-extendable to projective spanning trees. The proposed algorithms were applied for analyzing the possible variants of the syntactic subordination of nouns through prepositions .
In this paper we present an algorithm that generates $k$-noncrossing, $σ$-modular diagrams with uniform probability. A diagram is a labeled graph of degree $\le 1$ over $n$ vertices drawn in a horizontal line with arcs $(i,j)$ in the upper half-plane. A $k$-crossing in a diagram is a set of $k$ distinct arcs $(i_1, j_1), (i_2, j_2),\ldots,(i_k, j_k)$ with the property $i_1 < i_2 < \ldots < i_k < j_1 < j_2 < \ldots< j_k$. A diagram without any $k$-crossings is called a $k$-noncrossing diagram and a stack of length $σ$ is a maximal sequence $((i,j),(i+1,j-1),\dots,(i+(σ-1),j-(σ-1)))$. A diagram …
In a projective plane over a finite field, complete $(k,n)$-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with $k$ close to the order of the plane, is considered a hard problem. A natural candidate to be a small $(k,n)$-arc with few characters is the set $Ω(\mathcal{C})$ of the points of a plane curve $\mathcal{C}$ of degree $n$ (containing no linear components) such that some line meets $\mathcal{C}$ transversally in the plane, i.e. in $n$ pair…
Limits of combinatorial patchworking
本文通过构造扩张三角形拼图，实现了七次实平面代数曲线的121种实结构，并证明每种结构均可作为T曲线实现。
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