Maximal Arcs Research Articles

Disjoint maximal arcs in projective planes of order 16

Disjoint maximal arcs in projective planes of order 16

Near-MDS codes from maximal arcs in PG(2,q)

The singleton defect of an [n,k,d] linear code C is defined as s(C)=n−k+1−d. Codes with S(C)=0 are called maximum distance separable (MDS) codes, and codes with S(C)=S(C⊥)=1 are called near maximum distance separable (NMDS) codes. Both MDS codes and NMDS codes have good representations in finite projective geometry. MDS codes over Fq with length n and n-arcs in PG(k−1,q) are equivalent objects. When k=3, NMDS codes of length n are equivalent to (n,3)-arcs in PG(2,q). In this paper, we deal with the NMDS codes with dimension 3. By adding some suitable projective points in maximal arcs of PG(2,q), we can obtain two classes of (q+5,3)-arcs (or equivalently [q+5,3,q+2] NMDS codes) for any prime power q. We also determine the exact weight distribution and the locality of such NMDS codes and their duals. It turns out that the resultant NMDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes.

Disjoint sets in projective planes of small order

In this paper, results of a computer search for disjoint sets associated with maximal arcs and unitals in projective planes of order 16, and disjoint sets associated with unitals in projective planes of orders 9 and 25 are reported. It is shown that the number of pairs of disjoint unitals in planes of order 9 is exactly four, and new pairs and triples of disjoint degree 4 maximal arcs are shown to exist in some of the planes of order 16. New bounds on the number of 104-sets of type (4, 8) and 156-sets of type (8, 12) are achieved. A combinatorial method for finding new maximal arcs, new unitals, and new v-sets of type (m, n) is introduced. All disjoint sets found in this study are explicitly listed.

Extended cyclic codes, maximal arcs and ovoids

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 \([qt-q+t,3,qt-q]\), \(1<t<q\), q is a power of t, 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 simple presentations of Denniston maximal arcs in PG(2, q) and elliptic quadrics in PG(3, q).

On partial geometries arising from maximal arcs

The subject of this paper are partial geometries \(pg(s,t,\alpha )\) with parameters \(s=d(d'-1), \ t=d'(d-1), \ \alpha =(d-1)(d'-1)\), \(d, d' \ge 2\). 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 [28] and Wallis [34], with a single known exception of a partial geometry pg(4, 6, 3) found by Mathon [22] 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 covers the point set. Two parallel classes are called orthogonal if they share exactly one line. An upper bound on the maximum number of pairwise orthogonal parallel classes in a partial geometry G with parameters \(pg(d(d'-1),d'(d-1),(d-1)(d'-1))\) is proved, and it is shown that a necessary and sufficient condition for G to arise from a maximal arc of degree d or \(d'\) in a projective plane of order \(q=dd'\) is that both G and its dual geometry contain sets of pairwise orthogonal parallel classes that meet the upper bound. An alternative construction of Mathon’s partial geometry is presented, and the new necessary condition is used to demonstrate why this partial geometry is not associated with any maximal arc in the projective plane of order 8. The partial geometries associated with all known maximal arcs in projective planes of order 16 are classified up to isomorphism, and their parallel classes of lines and the 2-rank of their incidence matrices are computed. Based on these results, some open problems and conjectures are formulated.

Maximal Arcs, Codes, and New Links Between Projective Planes of Order 16

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 block spectrum of RNA pseudoknot structures.

In this paper we analyze the length-spectrum of blocks in [Formula: see text]-structures. [Formula: see text]-structures are a class of RNA pseudoknot structures that play a key role in the context of polynomial time RNA folding. A [Formula: see text]-structure is constructed by nesting and concatenating specific building components having topological genus at most [Formula: see text]. A block is a substructure enclosed by crossing maximal arcs with respect to the partial order induced by nesting. We show that, in uniformly generated [Formula: see text]-structures, there is a significant gap in this length-spectrum, i.e., there asymptotically almost surely exists a unique longest block of length at least [Formula: see text] and that with high probability any other block has finite length. For fixed [Formula: see text], we prove that the length of the complement of the longest block converges to a discrete limit law, and that the distribution of short blocks of given length tends to a negative binomial distribution in the limit of long sequences. We refine this analysis to the length spectrum of blocks of specific pseudoknot types, such as H-type and kissing hairpins. Our results generalize the rainbow spectrum on secondary structures by the first and third authors and are being put into context with the structural prediction of long non-coding RNAs.

The subfield codes of hyperoval and conic codes

Hyperovals in PG(2,GF(q)) with even q are maximal arcs and an interesting research topic in finite geometries and combinatorics. Hyperovals in PG(2,GF(q)) are equivalent to [q+2,3,q] MDS codes over GF(q), called hyperoval codes, in the sense that one can be constructed from the other. Ovals in PG(2,GF(q)) for odd q are equivalent to [q+1,3,q−1] MDS codes over GF(q), which are called oval codes. In this paper, we investigate the binary subfield codes of two families of hyperoval codes and the p-ary subfield codes of the conic codes. The weight distributions of these subfield codes and the parameters of their duals are determined. As a byproduct, we generalize one family of the binary subfield codes to the p-ary case and obtain its weight distribution. The codes presented in this paper are optimal or almost optimal in many cases. In addition, the parameters of these binary codes and p-ary codes seem new.

Maximal arcs and extended cyclic 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 $${\mathrm {PG}}(2,q)$$ , 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.

Linear codes from Denniston maximal arcs

In this paper we construct functional codes from Denniston maximal arcs. For $$q=2^{4\ell +2}$$ we obtain linear codes with parameters $$[(\sqrt{q}-1)(q+1),5,d]_q$$ where $$\lim _{q \rightarrow +\infty } d=(\sqrt{q}-1)q-\sqrt{q}$$ . 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.

Maximal arcs in projective planes of order 16 and related designs

Abstract 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.

On resolvable Steiner 2-designs and maximal arcs in projective planes

A combinatorial characterization of resolvable Steiner 2-(v, k, 1) designs embeddable as maximal arcs in a projective plane of order $$(v-k)/(k-1)$$(v-k)/(k-1) is proved, and a generalization of a conjecture by Andries Brouwer (Geometries and groups, Springer, Heidelberg, 1981) is formulated.

Maximal arcs and quasi-symmetric designs

We show that the construction of quasi-symmetric designs with parameters 2-$$(q^3, q^{2}(q-1)/2, q(q^3 -q^2 -2)/4)$$(q3,q2(q-1)/2,q(q3-q2-2)/4) and block intersection numbers $$q^{2}(q-2)/4$$q2(q-2)/4 and $$q^{2}(q-1)/4$$q2(q-1)/4 (where $$q \ge 4$$q?4 is a power of 2) given by Blokhuis and Haemers (J Stat Plan Inference 95:117---119, 2001) leads to exponential numbers of such designs. For $$q=4$$q=4, there are already at least 28,844 isomorphism classes.

Large Incidence-free Sets in Geometries

Let $\Pi = (P,L,I)$ denote a rank two geometry. In this paper, we are interested in the largest value of $|X||Y|$ where $X \subset P$ and $Y \subset L$ are sets such that $(X \times Y) \cap I = \emptyset$. Let $\alpha(\Pi)$ denote this value. We concentrate on the case where $P$ is the point set of $\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\mathsf{PG}(n,q)$. In the case that $\Pi$ is the projective plane $\mathsf{PG}(2,q)$, where $P$ is the set of points and $L$ is the set of lines of the projective plane, Haemers proved that maximal arcs in projective planes together with the set of lines not intersecting the maximal arc determine $\alpha(\mathsf{PG}(2,q))$ when $q$ is an even power of $2$. Therefore, in those cases,\[ \alpha(\Pi) = q(q - \sqrt{q} + 1)^2.\] We give both a short combinatorial proof and a linear algebraic proof of this result, and consider the analogous problem in generalized polygons. More generally, if $P$ is the point set of $\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\mathsf{PG}(n,q)$, where $1 \leq k \leq n - 1$, and $\Pi_q = (P,L,I)$, then we show as $q \rightarrow \infty$ that \[ \frac{1}{4}q^{(k + 2)(n - k)} \lesssim \alpha(\Pi) \lesssim q^{(k + 2)(n - k)}.\] The upper bounds are proved by combinatorial and spectral techniques. This leaves the open question as to the smallest possible value of $\alpha(\Pi)$ for each value of $k$. We prove that if for each $N \in \mathbb N$, $\Pi_N$ is a partial linear space with $N$ points and $N$ lines, then $\alpha(\Pi_N) \gtrsim \frac{1}{e}N^{3/2}$ as $N \rightarrow \infty$.

Partial flocks of the quadratic cone yielding Mathon maximal arcs

In [6] Hamilton and Thas (2006) 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 (see De Clerck et al. (2011) [3]). 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.

Singer 8-arcs of Mathon type in PG(2, 27)

De Clerck et al. (J Comb Theory, 2011) counted the number of non-isomorphic Mathon maximal arcs of degree-8 in PG(2, 2 h ), h ≠ 7 and prime. In this article we will show that in PG(2, 27) 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, 27) extend to two infinite families of Mathon arcs of degree-8 in PG(2, 2 k ), k odd and divisible by 7, while maintaining the nice property of admitting a Singer group.

A geometric approach to Mathon maximal arcs

In 1969 Denniston [3] 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 [8] 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.

An Algorithm for Constructing Some Maximal Arcs in PG(2, q 2)

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).

Algebraic curves and maximal arcs

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.

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

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.