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.
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 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…
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…
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…
In this paper, we study KM-arcs of type t, i.e. point sets of size q + t in PG(2, q) such that every line contains 0, 2 or t of its points. We use field reduction to give a different point of view on the class of translation arcs. Starting from a particular F2-linear set, called an i-club, we reconstruct the projective triads, the translation hyperovals as well as the translation arcs constructed by Korchmaros-Mazzocca, Gacs-Weiner and Limbupasiriporn. We show the KM-arcs of type q/4 recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a fa…
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.
Linear sets on the projective line have attracted a lot of attention because of their link with blocking sets, KM-arcs and rank-metric codes. In this paper, we study linear sets having two points of complementary weight, that is with two points for which the sum of their weights equals the rank of the linear set. As a special case, we study those linear sets having exactly two points of weight greater than one, by showing new examples and studying their equivalence issue. Also we determine some linearized polynomials defining the linear sets recently introduced by Jena and Van de Voorde (2021)…
Let $m$ be a positive integer, $q$ be a prime power, and $\mathrm{PG}(2,q)$ be the projective plane over the finite field $\mathbb F_q$. Finding complete $m$-arcs in $\mathrm{PG}(2,q)$ of size less than $q$ is a classical problem in finite geometry. In this paper we give a complete answer to this problem when $q$ is relatively large compared with $m$, explicitly constructing the smallest $m$-arcs in the literature so far for any $m\geq 8$. For any fixed $m$, our arcs $\mathcal A_{q,m}$ satisfy $|\mathcal A_{q,m}|-q\rightarrow -\infty$ as $q$ grows. To produce such $m$-arcs, we develop a Galois…
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…
In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $m\ge2$, a set of $m+1$ partitions of a set $Ω$, any $m$ of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if $m=2$), or generate a join-semilattice of dimension $m$ associated with a diagonal group over a base group $G$. In this paper we investigate what happens if we have $m+r$ partitions with $r\geq 2$, any $m$ of which are minimal elements of a Cartesian lattice. If $m=2$, this is just a set of mutually orthogonal Latin squares. We consider the cas…
