TY - JOUR
T1 - Boolean functions with MacWilliams duality
AU - Hyun, Jong Yoon
AU - Lee, Heisook
AU - Lee, Yoonjin
N1 - 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.
PY - 2014/8
Y1 - 2014/8
N2 - 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.
AB - 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.
KW - Bent function
KW - Formally self-dual function
KW - Formally self-dual pair
KW - MacWilliams duality
KW - MacWilliams-dual function
KW - Self-dual bent function
UR - http://www.scopus.com/inward/record.url?scp=84902182935&partnerID=8YFLogxK
U2 - 10.1007/s10623-012-9762-7
DO - 10.1007/s10623-012-9762-7
M3 - Article
AN - SCOPUS:84902182935
SN - 0925-1022
VL - 72
SP - 273
EP - 287
JO - Designs, Codes, and Cryptography
JF - Designs, Codes, and Cryptography
IS - 2
ER -