Leopardi-Paul.tex

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\newcommand{\FirstName}{Paul}
\newcommand{\LastName}{Leopardi}
\newcommand{\Uni}{University of Newcastle} % e.g. The University of Melbourne
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\newcommand{\TalkTitle}{Classifying bent functions by their Cayley graphs}
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\begin{TalkAbstract}
It is well known~\cite{BerC99} that if a bent function $f: \Z_2^{2m} \To \Z_2$ has $f(0)=0$, then it has a strongly regular Cayley graph whose parameters $(v_m,k_m,\lambda_m,\lambda_m)$ depend only on $m$:
\begin{align*}
(v_m,k_m,\lambda_m) &= (4^m, 2^{2 m - 1} \pm 2^{m-1}, 2^{2 m - 2} \pm 2^{m-1}).
\end{align*}
It is perhaps less well known that even if two such Cayley graphs have the same strongly regular graph parameters, they are not necessarily isomorphic.
This talk examines the concepts of \emph{Cayley equivalence} and \emph{extended Cayley equivalence} of bent functions, and compares these equivalence relations to the
better known concepts of affine equivalence and extended affine equivalence.
The relationship between two-weight codes, bent functions and strongly regular graphs is also touched on.
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\begin{thebibliography}{1}
\bibitem{BerC99}
A.~Bernasconi and B.~Codenotti.
\newblock Spectral analysis of {Boolean} functions as a graph eigenvalue
  problem.
\newblock {\em IEEE Transactions on Computers}, 48(3):345--351, (1999).
\end{thebibliography}
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