TY - JOUR

T1 - On metric complements and metric regularity in finite metric spaces

AU - Oblaukhov, A. K.

N1 - Funding Information: 1The work was carried out under the state contract of the Sobolev Institute of Mathematics (project no. 0314-2019-0017) and supported by RFBR (projects no. 18-07-01394, 19-31-90093) and Laboratory of Cryptography JetBrains Research. Publisher Copyright: © 2020 Tomsk State University. All rights reserved. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/9

Y1 - 2020/9

N2 - 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 b 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 b 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.

AB - 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 b 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 b 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.

KW - Bent function

KW - Covering radius

KW - Deep hole

KW - Linear code

KW - Metric complement

KW - Metrically regular set

KW - Reed - Muller code

KW - metric complement

KW - bent function

KW - linear code

KW - metrically regular set

KW - covering radius

KW - COVERING RADIUS

KW - deep hole

KW - REED-MULLER CODE

UR - http://www.scopus.com/inward/record.url?scp=85095714348&partnerID=8YFLogxK

U2 - 10.17223/20710410/49/3

DO - 10.17223/20710410/49/3

M3 - Article

AN - SCOPUS:85095714348

SP - 35

EP - 45

JO - Прикладная дискретная математика

JF - Прикладная дискретная математика

SN - 2071-0410

IS - 49

ER -