, Volume 72, Issue 2, pp 273-287

Boolean functions with MacWilliams duality

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Abstract

We introduce a new class of Boolean functions for which the MacWilliams duality holds, called MacWilliams-dual functions, by considering a dual notion on Boolean functions. By using the MacWilliams duality, we prove the Gleason-type theorem on MacWilliams-dual functions. We show that a collection of MacWilliams-dual functions contains all the bent functions and all formally self-dual functions. We also obtain the Pless power moments for MacWilliams-dual functions. Furthermore, as an application, we prove the nonexistence of bent functions in 2n variables with minimum degree nk for any nonnegative integer k and nN with some positive integer N under a certain condition.

Communicated by C. Carlet.