Twin Bent Functions and Clifford Algebras

Abstract

This paper examines a pair of bent functions on Z 2m 2 and their relationship to a necessary condition for the existence of an automorphism of an edge-coloured graph whose colours are defined by the properties of a canonical basis for the real representation of the Clifford algebra R m,m . Some other necessary conditions are also briefly examined. IntroductionA recent paper [11] constructs a sequence of edge-coloured graphs ∆ m (m 1) with two edge colours, and makes the conjecture that for m 1, there is an automorphism of ∆ m that swaps the two edge colours. This conjecture can be refined into the following question. Question 1.1. Consider the sequence of edge-coloured graphs ∆ m (m 1) as defined in [11], each with red subgraph ∆ m [−1], and blue subgraph ∆ m [1]. For which m 1 is there an automorphism of ∆ m that swaps the subgraphs ∆ m [−1] and ∆ m [1]?Note that the existence of such an automorphism automatically implies that the subgraphs ∆ m [−1] and ∆ m [1] are isomorphic.Considering that it is known that ∆ m [−1] is a strongly regular graph, a more general question can be asked concerning such graphs.

Twin bent functions, strongly regular Cayley graphs, and Hurwitz--Radon theory

“…This section sets out the remainder of the definitions and properties used in this paper. It is based on the previous papers [10,11] with additions.…”

“…The previous two papers [10,11] define two binary functions on Z 2m 2 , σ m and τ m , respectively. Their key properties are repeated below.…”

“…The author's subsequent paper on bent functions [11] refines Conjecture 1 into the following question. The main result of this paper, Theorem 1, leads to the resolution of these conjectures and this question.…”

“…The bent function σ m on Z 2m 2 is described in the first paper [10], on generalizations of Williamson's construction for Hadamard matrices. The bent function τ m on Z 2m 2 is described in the second paper [11], which investigates some of the properties of the two sequences of bent functions. In this second paper it is shown that the bent functions σ m and τ m both correspond to Hadamard difference sets with the same parameters (v m , k m , λ m , n m ) = (4 m , 2 2m−1 − 2 m−1 , 2 2m−2 − 2 m−1 , 2 2m−2 ), and that their corresponding Cayley graphs are both strongly regular with the same parameters (v m , k m , λ m , λ m ).…”

Gastineau-Hills’ Quasi-Clifford Algebras and Plug-In Constructions for Hadamard Matrices

“…Thanks to Jennifer Seberry, who supervised the PhD thesis of Humphrey Gastineau-Hills, and brought his work to the attention of the author. This work, including previous papers on this topic [12,13,14] began in 2007 while the author was a Visiting Fellow at the Australian National University; continued while the author was a Visiting Fellow and a Casual Academic at the University of Newcastle, Australia; and concluded while the author was an employee of the Australian Government in the Bureau of Meteorology, and also an Honorary Fellow of the University of Melbourne.…”