On metric complements and metric regularity in finite metric spaces

This review deals with the metric complements and metric regularity in the Boolean cube and in arbitrary finite metric spaces. Let $A$ be an arbitrary subset of a finite metric space $M$, and $\widehat{A}$ be the metric complement of $A$ — the set of all points of $M$ at the maximal possible distance from $A$. If the metric complement of the set $\widehat{A}$ coincides with $A$, then the set $A$ is called a metrically regular set. The problem of investigating metrically regular sets was posed by N. Tokareva in 2012 when studying metric properties of bent functions, which have important applications in cryptography and coding theory and are also one of the earliest examples of a metrically regular set. In this paper, main known problems and results concerning the metric regularity are overviewed, such as the problem of finding the largest and the smallest metrically regular sets, both in the general case and in the case of fixed covering radius, and the problem of obtaining metric complements and establishing metric regularity of linear codes. Results concerning metric regularity of partition sets of functions and Reed — Muller codes are presented.