An algebraic methodology for defining new metrics over two-dimensional signal spaces is presented in this work. We have mainly considered quadrature amplitude modulation (QAM) constellations which previously been modeled by quotient rings of Gaussian integers. The metric these constellations, based on the distance concept circulant graphs, one main contributions A detailed analysis some degree-four graphs has allowed us to detail weight distribution spaces. family perfect codes integers will...

Many current parallel computers are built around a torus interconnection network. Machines from Cray, HP, and IBM, among others, make use of this topology. In terms topological advantages, square (2D) or cubic (3D) tori would be the topologies choice. However, for different practical reasons, 2D 3D with number nodes per dimension have been used. These mixed-radix not edge symmetric, which translates into poor performance due to an unbalanced network resources. work, we analyze twisted that...

The search for perfect error-correcting codes has received intense interest since the seminal work by Hamming. Decades ago, Golomb and Welch studied Lee metric in multidimensional torus constellations. In this work, we focus our attention on a new class of four-dimensional signal spaces which include tori as subcases. Our constellations are modeled means Cayley graphs defined over quotient rings Lipschitz integers. Previously unexplored length one will be provided constructive way solving...

A complete family of Cayley graphs degree four, denoted as L-networks, is considered in this paper. L-networks are 2D mesh-based topologies with wrap-around connections. constitute a graph-based model which englobe many previously proposed interconnection networks. Some them have been extensively used the industry underlying topology for parallel and distributed computers different scales. Tori, twisted doubly tori, toroidal diagonal meshes, chordal rings, circulant are, among others,...

The interconnection network comprises a significant portion of the cost large parallel computers, both in economic terms and power consumption. Several previous proposals exploit large-radix routers to build scalable low-distance topologies with aim minimizing these costs. However, they fail consider potential unbalance utilization, which some cases results suboptimal designs. Based on an appropriate model, this paper advocates use networks based incidence graphs projective planes, broadly...

A construction of 2-quasi-perfect Lee codes is given over the space $\mathbb Z_p^n$ for $p$ prime, $p\equiv \pm 5\pmod{12}$ and $n=2[\frac{p}{4}]$. It known that there are infinitely many such primes. Golomb Welch conjectured perfect Lee-metric do not exist dimension $n\geq 3$ radius $r\geq 2$. This conjecture was proved to be true large radii as well low dimensions. The found very close perfect, which exhibits hardness conjecture. series computations show related graphs Ramanujan, could...

Many parallel computers use Tori interconnection networks. Machines from Cray, HP and IBM, among others, exploit these topologies. In order to maintain full network symmetry, 2D 3D must have the same number of nodes (k) per dimension resulting in square or cubic Nevertheless, for practical reasons, computer engineers designed built having a different dimension. These mixed-radix topologies are not edge-symmetric which translates into poor performance provoked by an unbalanced links. this...

The basis for designing error-correcting codes two dimensional signal sets is considered in this paper. Both, algebraic and graph-theoretical approaches are employed research establishing the fundamentals of these codes. We give a solution to t-dominating set problem subfamily degree four circulant graphs which directly provides perfect over Gaussian integers. In order show applicability our results, simple examples different coding schemes also presented

In datacenter networks, big scale, high performance and faulttolerance, low-cost, graceful expandability are pursued features. Recently, random regular as the Jellyfish, have been proposed for satisfying these stringent requirements. However, their completely unstructured design entails several drawbacks. As a related alternative, in this paper we propose Random Folded Clos (RFC) networks. They constitute compromise between total randomness maintaining some topological structure. it will be...

In this paper we present perfect codes for two-dimensional constellations derived from generalized Gaussian graphs, a family of graphs built over quotient rings integers. Using the distance, solve problem finding t-dominating sets and, then, build new these graphs. The well-known Lee can be viewed as particular subcase introduced in work

Cayley graphs over quotients of the quaternion integers are going to be used define a new metric four dimensional lattices. We will consider perfect 1-error correcting codes according this space. show that, in some cases, these lattices can represented as two-dimensional constellations, which allow us state relation between Lee and Lipschitz metric.

The problem of searching for perfect codes has attracted great attention since the paper by Golomb and Welch, in which existence these over Lee metric spaces was considered. Since are not very common, quasi-perfect is also interest. In this aspect, have been considered 2-D 3-D spaces. paper, constructive methods obtaining modeled means Gaussian Eisenstein-Jacobi integers given. obtained form ideals integer ring thus preserving property being geometrically uniform codes. Moreover, they able...

Circulant graphs have been deeply studied in technical literature. Midimew networks are a class of distance-related optimal circulant degree four which applications network engineering and coding theory. In this research, new layout for keeps the maximum link length under value /spl radic/5 is presented, considering unitary as subjacent mesh link's length. The most interesting sizes studied: dense quasidense cases, with bounded layouts both although proposed algorithm also valid other sizes....

In order to propose a new metric over QAM constellations, diagonal Gaussian graphs defined quotients of the integers are introduced in this paper. Distance properties constellations detailed by means vertex-to-vertex distribution family graphs. Moreover, perfect codes for considered. Finally, notable subgraphs studied which leads relate proposed other well-known graph-based metrics such as Lee distance.

Big scale, high performance and fault-tolerance, low-cost graceful expandability are pursued features in current datacenter networks (DCN). Although there have been many proposals for DCNs, most modern installations equipped with classical folded Clos networks. Recently, regular random topologies, as the Jellyfish, proposed DCNs. However, their completely unstructured nature entails serious design problems. In this paper we propose Random Folded (RFC) Hydra which interconnection between...

Torus networks of moderate degree have been widely used in the supercomputer industry. Tori are superb when for executing applications that require near-neighbor communications. Nevertheless, they not so good dealing with global Hence, typical 3D implementations evolved to 5D networks, among other reasons, reduce network distances. Most these big systems mixed-radix tori, which best option minimizing distances and efficiently using resources. This paper is focused on improving topological...

In this paper we consider perfect codes over two dimensional QAM-type constellations of any cardinal. Such are going to be modeled by L-graphs, which the two-dimensional family multidimensional circulants, defined. We show that Gaussian graphs, Lee graphs and Kronecker product cycles included in family. Therefore, our method obtain these lattice subsets is a generalization techniques for searching products cycles. addition, introduce some previously unreported codes.

In the late years many different interconnection networks have been used with two main tendencies. One is characterized by use of high-degree routers long wires while other uses much smaller degree. The latter rely on two-dimensional mesh and torus topologies shorter local links. This paper focuses doubling degree common 2D meshes tori still preserving an attractive layout for VLSI design. By adding a set diagonal links in one direction, are obtained. second links, eight built, named king...

This work attempts to compare size and cost of two network topologies proposed for large-radix routers: concentrated torus dragonflies. We study the scalability, fault tolerance each network. On average, we found that a can be cost-efficient option middle-range networks.

In this paper we present a distance-hereditary decomposition of optimal chordal rings 2k/sup 2/ nodes into set 2k nodes, where k is the diameter. All belonging to have same length and their diameter corresponds ring in which they are embedded. The members embedded non-disjoint preserve minimal routing original circulant graph. Besides its practical consequences, our research allows presentation these graphs as particular evolution traditional topology.