A survey of perfect codes

Heden, Olof

doi:10.3934/amc.2008.2.223

Cited by 56 publications

(47 citation statements)

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“…1-perfect codes in n-cubes (see Example 8 below) are actively studied; the best-investigated case is binary, see, for example, [12,22], but even in that case the problem of full characterization of such codes is far from being solved. This gives an interesting example of non-completely-regular codes whose code parameters guarantee that the code is a color of some perfect coloring.…”

Section: Perfect Colorings and Completely Regular Setsmentioning

confidence: 99%

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On weight distributions of perfect colorings and completely regular codes

Krotov

2010

Des. Codes Cryptogr.

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A vertex coloring of a graph is called "perfect" if for any two colors a and b, the number of the color-b neighbors of a color-a vertex x does not depend on the choice of x, that is, depends only on a and b (the corresponding partition of the vertex set is known as "equitable"). A set of vertices is called "completely regular" if the coloring according to the distance from this set is perfect. By the "weight distribution" of some coloring with respect to some set we mean the information about the number of vertices of every color at every distance from the set. We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular set (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the color composition over the set. For some partial cases of completely regular sets, we derive explicit formulas of weight distributions. Since any (other) completely regular set itself generates a perfect coloring, this gives universal formulas for calculating the weight distribution of any completely regular set from its parameters. In the case of Hamming graphs, we prove a very simple formula for the weight enumerator of an arbitrary perfect coloring.

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Section: Perfect Colorings and Completely Regular Setsmentioning

confidence: 99%

“…Let f 0 be the sum of f over the set R of zeroes ofg (12). Then the weight distribution of f with respect to R is…”

Section: Lemma 2 the Distance From A Vertex X To R Coincides With Thementioning

confidence: 99%

On weight distributions of perfect colorings and completely regular codes

Krotov

2010

Des. Codes Cryptogr.

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“…Since the beginning of coding theory in the late 1940s, perfect codes have been important objects of study in information theory; see the surveys [3,4] on perfect codes and related definitions in the classical setting. It is known in [5] that deciding whether a graph has a total perfect code is NP-complete.…”

Section: Introductionmentioning

confidence: 99%

Perfect codes in power graphs of finite groups

et al. 2017

Open Mathematics

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Abstract:The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes.

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“…The code C is a perfect 1-error correcting binary code, here for short a perfect code, if to any wordx of Z n 2 there is a unique wordc of C at distance at most one fromx. Perfect codes have been studied for more than 60 years, and there are many different constructions; see for example [7]. There is up to ''equivalence'' just one perfect code of lengths 3 and 7, respectively, but 5983 different perfect codes of length 15, see [17], and the number of different perfect codes of length n ≥ 31 is extremely large, see [13], although n + 1 always must be a power of 2.…”

Section: Introductionmentioning

confidence: 99%