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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.

MSC:
94B05 Linear codes, general
06E30 Boolean functions
94A60 Cryptography
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[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
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