Formal self duality

TL;DR

This paper explores the concept of formal self duality in finite abelian groups, providing new theoretical insights, properties, and examples, and establishing connections to dual codes and specific structured sets.

Contribution

It offers a precise definition of formally self dual sets, reduces them to primitive cases, and introduces new examples and properties, advancing the theoretical understanding of formal duality.

Findings

Formal self dual sets can be reduced to primitive cases.

Several properties of formally self dual sets are established.

New examples of formally self dual sets are constructed.

Abstract

We study the notion of formal self duality in finite abelian groups. Formal duality in finite abelian groups has been proposed by Cohn, Kumar, Reiher and Sch\"urmann. In this paper we give a precise definition of formally self dual sets and discuss results from the literature in this perspective. Also, we discuss the connection to formally dual codes. We prove that formally self dual sets can be reduced to primitive formally self dual sets similar to a previously known result on general formally dual sets. Furthermore, we describe several properties of formally self dual sets. Also, some new examples of formally self dual sets are presented within this paper. Lastly, we study formally self dual sets of the form $\{(x,F(x)) \ : \ x\in\mathbb{F}_{2^n}\}$ where $F$ is a vectorial Boolean function mapping $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2^n}$.