Enumeration of subsets with closedness in finite fields of characteristic 2

TL;DR

This paper investigates the spectrum of additive closedness (r-values) in subsets of finite fields of characteristic 2, providing enumeration and analysis that connect to various combinatorial structures.

Contribution

It introduces a method to enumerate and analyze r-values in subsets of finite fields of characteristic 2, linking these to known combinatorial configurations.

Findings

Enumeration of r-values in finite fields of characteristic 2

Representation of r-values as a spectrum of closedness

Connections to partial Steiner triple systems, sum-free sets, Sidon sets, and Schure triples

Abstract

The additive closedness in the subset of an additive group is termed as r-value. The nature of closedness in different subsets of fixed size is observed as a spectrum of r-values. We enumerate r-values of subsets in finite fields of characteristic 2 and represent them as the spectrum of values. Based on these values the subsets can be further studied as partial Steiner triple systems, sum-free sets, Sidon sets, and Schure triples.