Related papers: An algorithm for constructing some maximal arcs in…

We construct two-weight sets in PG$(3n-1,q)$, $n\geq2$ with the same weights as those that would arise from the blow-up of a maximal $q$-arc in PG$(2,q^n)$. The construction is of particular interest when $q$ is odd, as it is well known…

We give an algorithm to explicitly compute the largest subtree, in the local Bruhat-Tits tree for PSL_2(k), whose vertices correspond to orders containing a given suborder H, in terms of a set of generators for H. The shape of this subtree…

In this work we construct a new class of maximal partial spreads in $PG(4,q)$, that we call $q$-added maximal partial spreads. We obtain them by depriving a spread of a hyperplane of some lines and adding $q+1$ lines not of the hyperplane…

We give an algorithm to calculate the minimal and maximal genus of the orientable closed surface where a graph $G$ can be embedded. For this, we construct some special branched coverings of the 2-sphere. We apply this algorithm to calculate…

We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a…

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…

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 $m_r(2,q)$…

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…

Given two convex polygons $P$ and $Q$ with $n$ and $m$ edges, the maximum overlap problem is to find a translation of $P$ that maximizes the area of its intersection with $Q$. We give the first randomized algorithm for this problem with…

A k-arc in a Dearguesian projective plane whose secants meet some external line in k-1 points is said to be hyperfocused. Hyperfocused arcs are investigated in connection with a secret sharing scheme based on geometry due to Simmons. In…

We classify nets of conics in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent…

Three algorithms of Gram-Schmidt type are given that produce an orthogonal decomposition of finite $d$-dimensional symmetric, alternating, or Hermitian forms over division rings. The first uses $d^3/3+O(d^2)$ ring operations with very…

In the projective planes $\mathrm{PG}(2,q)$, we collect the smallest known sizes of complete arcs for the regions \begin{align*} &\mbox{all } q\le160001,~~ q \mbox{ prime power};\\ &Q_{4}=\{34 \mbox{ sporadic }q'\mbox{s in the interval…

Ovoids in $\PG(3, \gf(q))$ have been an interesting topic in coding theory, combinatorics, and finite geometry for a long time. So far only two families of ovoids are known. The first is the elliptic quadratics and the second is the Tits…

The existence problem for maximal sets of equiangular lines (or SICs) in complex Hilbert space of dimension $d$ remains largely open. In a previous publication (arXiv:2112.05552) we gave a conjectural algorithm for how to construct a SIC if…

We construct a set of points with $\Omega(n^2\log n)$ triples determining an angle $\theta$ whenever $\tan(\theta)$ is algebraic over $\mathbb{Q}$, matching the upper bound of Pach and Sharir. This improves upon the original construction,…

We construct an explicit, embedded degeneration of the general torus orbit closure in the maximal orthogonal Grassmannian OG(n,2n+1) into a union of Richardson varieties. In particular, we deduce a formula for the cohomology class of the…

In this short note, we give a novel algorithm for $O(1)$ round triangle counting in bounded arboricity graphs. Counting triangles in $O(1)$ rounds (exactly) is listed as one of the interesting remaining open problems in the recent survey of…

In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in $\mathbb{R}^r$ with angle $\arccos(\alpha)$ and gave a partial answer in the regime $r \leq 1/\alpha^2 - 2$. At the other extreme where…

We construct orthogonal arrays OA$_{\lambda} (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / \lambda$ is as large as possible; these…