Classifying bent functions
by Paul Leopardi
Abstract: Bent Boolean functions are fascinating and useful combinatorial objects, whose applications include coding theory and cryptography. In even dimensions, the bent functions are those Boolean functions that are at the maximum Hamming distance from any affine function. The number of bent functions explodes with dimension, and various concepts of equivalence are used to classify them. In 1999 Bernasconi and Codenotti noted that the Cayley graph of a bent function is strongly regular.
This talk describes the concept of extended Cayley equivalence of bent functions, explores the relationship between extended Cayley equivalence and extended affine equivalence of bent functions, and discusses some connections between bent functions, symmetric designs, linear codes, and strongly regular graphs. SageMath scripts and Cocalc worksheets are used to compute and display some of these relationships and connections, for bent functions up to dimension 8.
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