zbMATH — the first resource for mathematics

MacWilliams duality and a Gleason-type theorem on self-dual bent functions. (English) Zbl 1259.94071
Summary: We prove that the MacWilliams duality holds for bent functions. It enables us to derive the concept of formally self-dual Boolean functions with respect to their near weight enumerators. Using this concept, we prove a Gleason-type theorem for self-dual bent functions. As an application, we provide the total number of (self-dual) bent functions in two and four variables obtained from formally self-dual Boolean functions.

94B05 Linear codes, general
06E30 Boolean functions
94A60 Cryptography
Full Text: DOI
[1] Carlet C.: Boolean Functions for Cryptography and Error Correcting Codes in Boolean Methods and Models. Cambridge University Press, Cambridge (to appear). · Zbl 1209.94035
[2] Carlet C., Danielsen L.E., Parker M.G., Sole P.: Self-dual bent functions. Int. J. Inform. Coding Theory 1(4), 384–399 (2010) · Zbl 1204.94118 · doi:10.1504/IJICOT.2010.032864
[3] Hyun J.Y.: Generalized MacWilliams identities and their applications to perfect binary codes. Des. Codes Cryptogr 50(2), 173–185 (2009) · Zbl 1237.94125 · doi:10.1007/s10623-008-9222-6
[4] Huffman W.G., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003) · Zbl 1099.94030
[5] MacWilliams F.J., Sloane N.J.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1998) · Zbl 0369.94008
[6] Rothaus O.S.: On bent functions. J. Combin. Theory Ser. A 20(3), 300–305 (1976) · Zbl 0336.12012 · doi:10.1016/0097-3165(76)90024-8
[7] Yin T., Alexander P., Tao F.: Strongly regular graphs associated with ternary bent functions. J. Combin. Theory Ser. A 117(6), 668–682 (2010) · Zbl 1267.05300 · doi:10.1016/j.jcta.2009.05.003
[8] van Lint J.H.: Kerdock codes and preparata codes. In: Proceedings of the Fourteenth Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL. Congr. Numer. 39, pp. 2541 (1983). · Zbl 0549.94027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.