Related papers: Elusive Codes in Hamming Graphs
Orbits of graphs under local complementation (LC) and edge local complementation (ELC) have been studied in several different contexts. For instance, there are connections between orbits of graphs and error-correcting codes. We define a new…
Constant composition codes have been proposed as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication. In particular, a certain class of constant composition codes called…
A graph property is elusive (or evasive) if any algorithm testing it by asking questions of the form ''Is there an edge between vertices x and y?'' must, in the worst case, examine all pairs of vertices. Elusiveness for infinite vertex sets…
Binary self-dual codes with large minimum distances, such as the extended Hamming code and the Golay code, are fascinating objects in the coding theory. They are closely related to sporadic simple groups, lattices and invariant theory. A…
The completely regular codes in Hamming graphs have a high degree of combinatorial symmetry and have attracted a lot of interest since their introduction in 1973 by Delsarte. This paper studies the subfamily of completely transitive codes,…
We study ensembles of codes on graphs (generalized low-density parity-check, or LDPC codes) constructed from random graphs and fixed local constrained codes, and their extension to codes on hypergraphs. It is known that the average minimum…
Let $X$ be a subgroup of the full automorphism group of the Hamming graph $H(m,q)$, and $C$ a subset of the vertices of the Hamming graph. We say that $C$ is an \emph{$(X,2)$-neighbour transitive code} if $X$ is transitive on $C$, as well…
This is a chapter in a forthcoming book on completely regular codes in distance regular graphs. The chapter provides an overview, and some original results, on codes in distance regular graphs which admit symmetries via a permutation group…
An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We study the edge-identifying code problem, i.e. the identifying code…
A detection system, modeled in a graph, is composed of "detectors" positioned at a subset of vertices in order to uniquely locate an ``intruder" at any vertex. \emph{Identifying codes} use detectors that can sense the presence or absence of…
An equidistant code is a code in the Hamming space such that two distinct codewords have the same Hamming distance. This paper investigates the bounds for equidistant codes in Hamming spaces.
The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect…
Pairing-based cryptographic schemes require so-called pairing-friendly elliptic curves, which have special properties. The set of pairing-friendly elliptic curves that are generated by given polynomials form a complete family. Although a…
We investigate the list decodability of symbol-pair codes in the present paper. Firstly, we show that list decodability of every symbol-pair code does not exceed the Gilbert-Varshamov bound. On the other hand, we are able to prove that with…
A property of n-vertex graphs is called evasive if every algorithm testing this property by asking questions of the form "is there an edge between vertices u and v" requires, in the worst case, to ask about all pairs of vertices. Most…
An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from all other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of…
Building on previous results of Xing, we give new lower bounds on the rate of intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry, although nothing beyond the basic theory of linear systems on…
A large family of linear codes with flexible parameters from almost bent functions and perfect nonlinear functions are constructed and their parameters are determined. Some constructed linear codes and their related codes are optimal in the…
Motivated by systems where the information is represented by a graph, such as neural networks, associative memories, and distributed systems, we present in this work a new class of codes, called codes over graphs. Under this paradigm, the…
In two previous papers we constructed new families of completely regular codes by concatenation methods. Here we determine cases in which the new codes are completely transitive. For these cases we also find the automorphism groups of such…