Quotients of Gaussian graphs and their application to perfect codes (original) (raw)
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2013
Motivated by a problem in computer architecture we introduce a notion of the perfect distance-dominating set, PDDS, in a graph. PDDSs constitute a generalization of perfect Lee codes, diameter perfect codes, as well as other codes and dominating sets. In this paper we initiate a systematic study of PDDSs. PDDSs related to the application will be constructed and the non-existence of some PDDSs will be shown. In addition, an extension of the long-standing Golomb-Welch conjecture, in terms of PDDS, will be stated. We note that all constructed PDDSs are lattice-like which is a very important feature from the practical point of view as in this case decoding algorithms tend to be much simpler.
Graph Theoretic Methods in Coding Theory
Classical, Semi-classical and Quantum Noise, 2011
This paper is a tutorial on the application of graph theoretic techniques in classical coding theory. A fundamental problem in coding theory is to determine the maximum size of a code satisfying a given minimum Hamming distance. This problem is thought to be extremely hard and still not completely solved. In addition to a number of closed form expressions for special cases and some numerical results, several relevant bounds have been derived over the years.
0 SQS-graphs of extended 1-perfect codes
2016
A binary extended 1-perfect code C folds over its kernel via the Steiner quadruple systems associated with its codewords. The resulting folding, proposed as a graph invariant for C, distinguishes among the 361 nonlinear codes C of kernel dimension κ with 9 ≥ κ ≥ 5 obtained via Solov'eva-Phelps doubling construction. Each of the 361 resulting graphs has most of its nonloop edges expressible in terms of the lexicographically disjoint quarters of the products of the components of two of the ten 1-perfect partitions of length 8 classified by Phelps, and loops mostly expressible in terms of the lines of the Fano plane.
SQS-graphs of extended 1-perfect codes
2009
A binary extended 1-perfect code C folds over its kernel via the Steiner quadruple systems associated with its codewords. The resulting folding, proposed as a graph invariant for C, distinguishes among the 361 nonlinear codes C of kernel dimension κ with 9 ≥ κ ≥ 5 obtained via Solov'eva-Phelps doubling construction. Each of the 361 resulting graphs has most of its nonloop edges expressible in terms of the lexicographically disjoint quarters of the products of the components of two of the ten 1-perfect partitions of length 8 classified by Phelps, and loops mostly expressible in terms of the lines of the Fano plane.
Perfect Codes for Metrics Induced by Circulant Graphs
IEEE Transactions on Information Theory, 2000
... In this paper, we adopt both, algebraic and graph-theoretical approaches to obtain a suitable metric over lattice constellations based on the Gaussian inte-gers, that corresponds to the ... Then, we will show how circulant graphs can be used for designing perfect codes in which ...
Isodual and self-dual codes from graphs
Algebra and Discrete Mathematics, 2021
Binary linear codes are constructed from graphs, in particular, by the generator matrix [In|A] where A is the adjacency matrix of a graph on n vertices. A combinatorial interpretation of the minimum distance of such codes is given. We also present graph theoretic conditions for such linear codes to be Type I and Type II self-dual. Several examples of binary linear codes produced by well-known graph classes are given.
Total perfect codes in Cayley graphs
Designs, Codes and Cryptography, 2016
A total perfect code in a graph Γ is a subset C of V (Γ) such that every vertex of Γ is adjacent to exactly one vertex in C. We give necessary and sufficient conditions for a conjugation-closed subset of a group to be a total perfect code in a Cayley graph of the group. As an application we show that a Cayley graph on an elementary abelian 2-group admits a total perfect code if and only if its degree is a power of 2. We also obtain necessary conditions for a Cayley graph of a group with connection set closed under conjugation to admit a total perfect code.
Graphs, Tessellations, and Perfect Codes on Flat Tori
IEEE Transactions on Information Theory, 2004
Quadrature amplitude modulation (QAM)-like signal sets are considered in this paper as coset constellations placed on regular graphs on surfaces known as flat tori. Such signal sets can be related to spherical, block, and trellis codes and may be viewed as geometrically uniform (GU) in the graph metric in a sense that extends the concept introduced by Forney [13]. Homogeneous signal sets of any order can then be labeled by a cyclic group, induced by translations on the Euclidean plane. We construct classes of perfect codes on square graphs including Lee spaces, and on hexagonal and triangular graphs, all on flat tori. Extension of this approach to higher dimensions is also considered.
Nearly perfect codes in distance-regular graphs
Discrete Mathematics, 1976
The idea of d nearly perfect code in a vector space over a binary field is gencralised to the class of distance-rcguiar graphs, A necessary condition for the existence of il ncarfy perfect code in a distance-regular graph is obtained, and it is shown thnt this condition impk the andogous rcnrlt in the classical binary case.