Abstract
Let A be an arbitrary subset of the Boolean cube, and \(\widehat {A}\) be the set of all vectors of the Boolean cube, which are at the maximal possible distance from the set A. If the set of all vectors at the maximal distance from \(\widehat {A}\) coincides with A, then the set A is called a metrically regular set. The problem of investigating metrically regular sets appears when studying bent functions, which have important applications in cryptography and coding theory. In this work a special subclass of strongly metrically regular subsets of the Boolean cube is studied. An iterative construction of strongly metrically regular sets is obtained. The formula for the number of sets which can be obtained via this construction is derived. Constructions for two families of large metrically regular sets are presented. Exact sizes of sets from these families are calculated. These sizes give us the best lower bound on sizes of largest metrically regular subsets of the Boolean cube.
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Acknowledgments
The author was supported by the Russian Foundation for Basic Research (projects no. 17-41-543364, 18-31-00479), by the Russian Ministry of Science and Education (the 5-100 Excellence Programme and the Project no. 1.12875.2018/12.1), by the program of fundamental scientific researches of the SB RAS no.I.5.1. (project no. 0314-2016-0017).
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Oblaukhov, A. A lower bound on the size of the largest metrically regular subset of the Boolean cube. Cryptogr. Commun. 11, 777–791 (2019). https://doi.org/10.1007/s12095-018-0326-1
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DOI: https://doi.org/10.1007/s12095-018-0326-1