arXiv subjects

Natalie Dodson

Publications and source records attributed to Natalie Dodson.

3 recordsLinked to original sources

arXiv2023

No-Three-in-a-$Θ:$ Variations on the No-Three-in-a-Line Problem

We pose a natural generalization to the well-studied and difficult no-three-in-a-line problem: How many points can be chosen on an $n \times n$ grid such that no three of them form an angle of $θ$? In this paper, we classify which angles yield nontrivial problems, noting that some angles appear in surprising configurations on the grid. We prove a lower bound of $2n$ points for angles $θ$ such that $135^\circ \leq θ< 180^\circ$, and further explore the case $θ= 135^\circ$, utilizing geometric properties of the grid to prove an upper bound of $3n - 2$ points. Lastly, we generalize the proof strategy used in proving the upper bound for $θ= 135^\circ$ to provide a general upper bound for all angles.

math.CO↗

arXiv2023

Universality for graphs with bounded density

A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an $\mathcal{H}$-universal graph can have. With the aim of unifying a number of recent results, we consider a family of graphs with bounded density. In particular, we construct a graph with $O_d\left( n^{2 - 1/(\lceil d \rceil + 1)} \right)$ edges which contains every $n$-vertex graph with density at most $d \in \mathbb{Q}$ ($d \ge 1$), which is close to a lower bound $\Omega(n^{2 - 1/d - o(1)})$ obtained by counting lifts of a carefully chosen (small) graph. When restricting the maximum degree of such graphs to be constant, we obtain a near-optimal universality. If we further assume $d \in \mathbb{N}$, we get an asymptotically optimal construction.

math.CO↗

arXiv2022

Improving the minimum distance bound of Trace Goppa codes

In this article we prove that a class of Goppa codes whose Goppa polynomial is of the form $g(x) = x + x^q + \cdots + x^{q^{m-1}}$ where $m \geq 3$ (i.e. $g(x)$ is a trace polynomial from a field extension of degree $m \geq 3$) has a better minimum distance than what the Goppa bound $d \geq 2deg(g(x))+1$ implies. Our improvement is based on finding another Goppa polynomial $h$ such that $C(L,g) = C(M, h)$ but $deg(h) > deg(g)$. This is a significant improvement over Trace Goppa codes over quadratic field extensions (i.e. the case $m = 2$), as the Goppa bound for the quadratic case is sharp.

cs.IT↗