On subgroup perfect codes in Cayley graphs

Asian-European Journal of Mathematics

The relative Cayley graph of a group [Formula: see text] with respect to its proper subgroup [Formula: see text] is a graph whose vertices are elements of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] for some [Formula: see text], where [Formula: see text] is an inverse-closed subset of [Formula: see text]. We study the relative Cayley graphs and, among other results, we discuss on their connectivity and forbidden structures, and compute some of their important numerical invariants.

Designs, Codes and …, 2001

The concept of diameter perfect codes, which seems to be a natural generalization of perfect codes (codes attaining the sphere-packing bound) is introduced. This was motivated by the "code-anticode" bound of Delsarte in distance regular graphs. This bound in conjunction with the recent complete solutions of diametric problems in the Hamming graph H q (n) and the Johnson graph J (n, k) gives a sharpening of the sphere-packing bound. Some necessary conditions for the existence of diameter perfect codes are given. In the Hamming graph all diameter perfect codes over alphabets of prime power size are characterized. The problem of tiling of the vertex set of J (n, k) with caps (and maximal anticodes) is also examined.

Algebra, Codes and Cryptology, 2019

Mathematical objects in this paper are group codes. In the first part of the paper we present a survey with some of the main results about group codes, mainly the existence of group codes that are not abelian group codes, the minimal length and the minimal dimension of such codes and the existence of a non-abelian group code that has better parameters than any abelian group code. In particular, in a previous paper [1], we have shown that the minimal dimension of a group code that is not abelian group code is 4. However, all known examples of group codes of dimension 4 that are non-abelian group codes are constructed using groups that are not p-groups. We do not know if such codes exist for the case of p-groups, but in the second part of this paper we prove that, under some restrictions on the base field, all four-dimensional G-codes for an arbitrary finite p-group G are abelian.

European Journal of Combinatorics, 1999

For a subset S of a group G such that 1 / ∈ S and S = S −1 , the associated Cayley graph Cay(G, S) is the graph with vertex set G such that {x, y} is an edge if and only if yx −1 ∈ S. Each σ ∈ Aut(G) induces an isomorphism from Cay(G, S) to the Cayley graph Cay(G, S σ). For a positive integer m, the group G is called an m-CI-group if, for all Cayley subsets S of size at most m, whenever Cay(G, S) ∼ = Cay(G, T) there is an element σ ∈ Aut(G) such that S σ = T. It is shown that if G is an m-CI-group for some m ≥ 4, then G = U × V , where (|U |, |V |) = 1, U is abelian, and V belongs to an explicitly determined list of groups. Moreover, Sylow subgroups of such groups satisfy some very restrictive conditions. This classification yields, as corollaries, improvements of results of Babai and Frankl. We note that our classification relies on the finite simple group classification.

Designs, Codes and Cryptography, 2009

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism FG → F n which maps G to the standard basis of F n. Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space F n , which does not assume an "a priori" group algebra structure on F n. As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed-Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes.

IEEE Transactions on Information Theory, 2000

... Hence, the cases in which the elements of can be arranged as a two-dimensional set will be considered according to the fol-lowing definition. Definition 17: Let . ... Page 6. MARTÍNEZ et al.:PERFECT CODES FROM CAYLEY GRAPHS OVER LIPSCHITZ INTEGERS ...

Communications in Algebra, 2020

For a character v of a finite group G, the number codðvÞ ¼ jG : kerðvÞj=vð1Þ is called the codegree of v. In this paper we show that if G is nonsolvable, then the codegree graph CðGÞ has a triangle. Also by using this result we show that if G has one composite codegree, then G is solvable and jcodðGÞj 5: ARTICLE HISTORY

Graphs and Combinatorics, 2019

In this paper, we define meta-Cayley graphs on dihedral groups. We fully determine the automorphism groups of the constructed graphs in question. Further, we prove that some of the graphs that we have constructed do not admit subgroups which act regularly on their vertex set; thus proving that they cannot be represented as Cayley graphs on groups.

International Journal of Applied Sciences and Smart Technologies, 2021

Let G be a fnite group with the set of subgroups of G denoted by S(G), then the subgroup graphs of G denoted by T(G) is a graph which set of vertices is S(G) such that two vertices H, K in S(G) (H not equal to K)are adjacent if either H is a subgroup of K or K is a subgroup of H. In this paper, we introduce the Subgroup graphs T associated with G. We investigate some algebraic properties and combinatorial structures of Subgroup graph T(G) and obtain that the subgroup graph T(G) of G is never bipartite. Further, we show isomorphism and homomorphism of the Subgroup graphs of finite groups. Let be a finite group with the set of subgroups of denoted by , then the subgroup graphs of denoted by is a graph which set of vertices is such that two vertices , are adjacent if either is a subgroup of or is a subgroup of . In this paper, we introduce the Subgroup graphs associated with . We investigate some algebraic properties and combinatorial structures of Subgroup graph and obtain that the su...

Proyecciones (Antofagasta), 2021

Let G be a group and S be a subset of G such that e ∉ S and S−1 ⊆ S. Then Cay(G, S) is a simple undirected Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy−1 ∈ S. The size of subset S is called the valency of Cay(G, S). In this paper, we determined the structure of all Cay(D2n, S), where D2n is a dihedral group of order 2n, n ≥ 3 and |S| = 1, 2 or 3.