Undergraduate honors thesis defense
Fri, 3 May, 2024
2:00pm - 3:00pm
Title: On m-general sets in AG(k,q) and PG(k,q): Sidon sets, projective arcs and error-correcting codes
Speaker: Tim Neumann
Date and Time: Friday, May 3, 2024, 2pm
Place: Rome 205
Abstract: A set of points in a geometric space is called m-general if any of its m-subsets is in general position. Such sets are inherently fascinating mathematical objects and also bear useful connections to coding theory.
In this thesis defense, we firstly examine 4-general sets in affine space over a finite field, where they are related to the well-studied Sidon sets from additive combinatorics. Working on the problem of finding large Sidon sets, we prove an improved lower bound on the maximal size of such sets and present novel computational results and methods. We then consider m-general sets over projective space in a synopsis of their relationship to error-correcting codes, illustrating how projective arcs give rise to maximum distance separable codes.
Throughout, we hope to create an appreciation of how seemingly unrelated mathematical objects (such as Sidon sets and error-correcting codes) are united by the notion of m-general sets.