Classifying bent functions by their Cayley graphs

TL;DR

This paper investigates the classification of bent functions through their Cayley graphs, introducing extended Cayley equivalence and exploring its connections to designs, codes, and affine equivalence, supported by computational tools.

Contribution

It introduces the concept of extended Cayley equivalence for bent functions and analyzes its relationship with extended affine equivalence, using computational methods.

Findings

Extended Cayley equivalence differs from extended affine equivalence.

Computations for bent functions up to dimension 8 reveal structural relationships.

SageMath and CoCalc tools facilitate the analysis of these relationships.

Abstract

In 1999 Bernasconi and Codenotti noted that the Cayley graph of a bent function is strongly regular. This paper describes the concept of extended Cayley equivalence of bent functions, discusses some connections between bent functions, designs, and codes, and explores the relationship between extended Cayley equivalence and extended affine equivalence. SageMath scripts and CoCalc worksheets are used to compute and display some of these relationships, for bent functions up to dimension 8.