A lower bound on the size of the largest metrically regular subset of the Boolean cube.

*(English)*Zbl 1456.94102Summary: 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.

##### MSC:

94A60 | Cryptography |

94D10 | Boolean functions |

68R01 | General topics of discrete mathematics in relation to computer science |

05C35 | Extremal problems in graph theory |

##### Keywords:

metrically regular set; metric complement; Boolean cube; bent function; strongly metrically regular set
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\textit{A. Oblaukhov}, Cryptogr. Commun. 11, No. 4, 777--791 (2019; Zbl 1456.94102)

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