On metric complements and metric regularity in finite metric spaces

A. K. Oblaukhov

Prikladnaâ diskretnaâ matematika, no. 3 (2020), pp. 35-45.

Résumé

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.

Détail

Keywords: metrically regular set, metric complement, covering radius, bent function, deep hole, Reed — Muller code, linear code.

Bibliographie

[1] Tokareva N., “Duality between bent functions and affine functions”, Discrete Mathematics, 312:3 (2012), 666–670 | DOI | MR | Zbl

[2] Tokareva N., Bent Functions: Results and Applications to Cryptography, Academic Press, 2015, 220 pp. | MR | Zbl

[3] Rothaus O. S., “On “bent” functions.”, J. Combin. Theory. Ser. A, 20:3 (1976), 300–305 | DOI | MR | Zbl

[4] Cusick T. W., Stănică P., Cryptographic Boolean Functions and Applications, Academic Press, 2017, 288 pp. | MR | Zbl

[5] Mesnager S., Bent Functions: Fundamentals and Results, Springer International Publishing, 2016, 544 pp. | MR | Zbl

[6] Tokareva N. N., “The group of automorphisms of the set of bent functions”, Discrete Math. Appl., 20:5–6 (2010), 655–664 | MR | Zbl

[7] Stănică P., Sasao T., Butler J. T., “Distance duality on some classes of Boolean functions.”, J. Combin. Math. Combin. Computing, 107 (2018), 181–198 | MR

[8] Oblaukhov A. K., “Maximal metrically regular sets”, Siberian Electronic Math. Reports, 15 (2018), 1842–1849 | MR | Zbl

[9] Oblaukhov A., “A lower bound on the size of the largest metrically regular subset of the Boolean cube”, Cryptogr. Commun., 11:4 (2019), 777–791 | DOI | MR | Zbl

[10] Oblaukhov A. K., “Metric complements to subspaces in the Boolean cube”, J. Appl. Industr. Math., 10:3 (2016), 397–403 | DOI | MR | Zbl

[11] Oblaukhov A., On metric regularity of Reed — Muller codes, 2020, arXiv: abs/1912.10811

[12] Cohen G., Honkala I., Litsyn S., Lobstein A., Covering Codes, North-Holland Mathematical Library, 54, Elsevier, 1997 | MR | Zbl

[13] Cohen G., Lobstein A., Sloane N., “Further results on the covering radius of codes.”, IEEE Trans. Inform. Theory, 32:5 (1986), 680–694 | DOI | MR | Zbl

[14] Graham R. L., Sloane N., “On the covering radius of codes.”, IEEE Trans. Inform. Theory, 31:3 (1985), 385–401 | DOI | MR | Zbl

[15] Tokareva N., Gorodilova A., Agievich S., et al., “Mathematical methods in solutions of the problems presented at the Third International Students' Olympiad in Cryptography”, Prikladnaya Diskretnaya Matematika, 2018, no. 40, 34–58 | DOI | MR

[16] Neumaier A., “Completely regular codes”, Discrete Math., 106 (1992), 353–360 | DOI | MR | Zbl

[17] Kutsenko A., “Metrical properties of self-dual bent functions”, Designs, Codes Cryptography, 88:1 (2020), 201–222 | DOI | MR | Zbl

[18] Kolomeec N. A., Pavlov A. V., “Properties of bent functions which are at minimal distance from each other”, Prikladnaya Diskretnaya Matematika, 2009, no. 4(6), 5–20 (in Russian)

[19] Kolomeets N. A., “Enumeration of the bent functions of least deviation from a quadratic bent function”, J. Appl. Industr. Math., 6:3 (2012), 306–317 | DOI | MR | Zbl

[20] Kolomeec N. A., “Upper bound on the number of bent functions which are at distance $2^k$ from an arbitrary bent function”, Prikladnaya Diskretnaya Matematika, 2014, no. 3(25), 28–39 (in Russian) | DOI

[21] Kolomeec N., “The graph of minimal distances of bent functions and its properties”, Designs, Codes Cryptography, 85:3 (2017), 395–410 | DOI | MR | Zbl

[22] Berlekamp E. and Welch N., “Weight distributions of the cosets of the $(32, 6)$ Reed — Muller code”, IEEE Trans. Inform. Theory, 18:1 (1972), 203–207 | DOI | MR | Zbl

[23] McLoughlin A. M., “The covering radius of the $(m-3)$-rd order Reed — Muller codes and a lower bound on the $(m-4)$-th order Reed Muller codes”, SIAM J. Appl. Math., 37:2 (1979), 419–422 | DOI | MR | Zbl

[24] Schatz J., “The second order Reed — Muller code of length $64$ has covering radius $18$”, IEEE Trans. Inform. Theory, 17:4 (1981), 529–530 | DOI | MR

[25] Mykkeltveit J., “The covering radius of the $(128, 8)$ Reed — Muller code is $56$”, IEEE Trans. Inform. Theory, 26:3 (1980), 359–362 | DOI | MR | Zbl

[26] Hou X. D., “Covering radius of the Reed — Muller code $R(1, 7)$ — a simpler proof”, J. Combin. Theory. Ser. A, 74:2 (1996), 337–341 | DOI | MR | Zbl

[27] Wang Q., “The covering radius of the Reed — Muller code $RM(2,7)$ is $40$”, Discrete Math., 342:12 (2019), 111625 | DOI | MR | Zbl