Boolean functions with MacWilliams duality

Jong Yoon Hyun, Heisook Lee, Yoonjin Lee

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 n-k for any nonnegative integer k and n ≥ N with some positive integer N under a certain condition.

Original languageEnglish
Pages (from-to)273-287
Number of pages15
JournalDesigns, Codes, and Cryptography
Issue number2
StatePublished - Aug 2014

Bibliographical note

Funding Information:
Acknowledgments The first named author was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korea government(MEST)(No. 2011-0010328), and the third named author is a corresponding author and was supported by Priority Research Centers Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(No. 2012-0006691) and by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST)(No. 2012-0005432). The authors would like to express sincere gratitude to anonymous referees for valuable comments and suggestions, which improved the exposition of the article. We also thank Dr. Hyun Jin Kim for his help in finishing Table 3.


  • Bent function
  • Formally self-dual function
  • Formally self-dual pair
  • MacWilliams duality
  • MacWilliams-dual function
  • Self-dual bent function


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