Related papers: Arithmetic completely regular codes
We consider the problem of existence of perfect $2$-colorings (equitable $2$-partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we…
The maximum independent sets in the Doob graphs D(m,n) are analogs of the distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove the characterization of these sets stating that every such set is semilinear or…
In this paper, we show how certain three-class association schemes and orthogonal arrays give rise to partial geometric designs. We also investigate the connections between partial geometric designs and certain regular graphs having three…
We give a complete characterization of simple graphs whose adjacency matrices generate binary linear complementary dual (LCD) codes. In particular, we completely characterize a distance-regular graph which yields an LCD code in terms of the…
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…
Given a graph $\Gamma$, a perfect code in $\Gamma$ is an independent set $C$ of vertices of $\Gamma$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code in $\Gamma$ is a set $C$ of vertices…
The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system, whose hamiltonian is identical to the adjacency matrix of a…
We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method we construct self-orthogonal codes from some distance-regular graphs. Some of the obtained codes are optimal.…
A perfect code in a graph is an independent set of the graph such that every vertex outside the set is adjacent to exactly one vertex in the set. A circulant graph is a Cayley graph of a cyclic group. In this paper we study perfect codes in…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex $u$ induces the same regular partition around $u$, where all vertices of each cell…
In this paper we provide an asymptotic expansion for the number of independent sets in a general class of regular, bipartite graphs satisfying some vertex-expansion properties, extending results of Jenssen and Perkins on the hypercube and…
The Lloyd Theorem of (Sol\'e, 1989) is combined with the Schwartz-Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank…
Constructions of woven graph codes based on constituent block and convolutional codes are studied. It is shown that within the random ensemble of such codes based on $s$-partite, $s$-uniform hypergraphs, where $s$ depends only on the code…
Highly-regular graphs can be regarded as a combinatorial generalization of distance-regular graphs. From this standpoint, we study combinatorial aspects of highly-regular graphs. As a result, we give the following three main results in this…
The sum-rank metric arises as an algebraic approach for coding in MIMO block-fading channels and multishot network coding. Codes designed in the sum-rank metric have raised interest in applications such as streaming codes, robust coded…
We show that the large Cartesian powers of any graph have log-concave valencies with respect to a ffxed vertex. We show that the series of valencies of distance regular graphs is log-concave, thus improving on a result of (Taylor,…
A subset \( C \) of the vertex set \( V \) of a graph \( \Gamma = (V,E) \) is termed an $(r,s)$-regular set if each vertex in \( C \) is adjacent to exactly \( r \) other vertices in \( C \), while each vertex not in \( C \) is adjacent to…
The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among…
We consider extended $1$-perfect codes in Hamming graphs $H(n,q)$. Such nontrivial codes are known only when $n=2^k$, $k\geq 1$, $q=2$, or $n=q+2$, $q=2^m$, $m\geq 1$. Recently, Bespalov proved nonexistence of extended $1$-perfect codes for…