Packing Radius Research Articles

Block codes on pomset metric

Given a regular multiset [Formula: see text] on [Formula: see text], a partial order [Formula: see text] on [Formula: see text], and a label map [Formula: see text] defined by [Formula: see text] with [Formula: see text], we define a pomset block metric [Formula: see text] on the direct sum [Formula: see text] of [Formula: see text] based on the pomset [Formula: see text]. The pomset block metric extends the classical pomset metric introduced by Sudha and Selvaraj and generalizes the poset block metric introduced by Alves et al. over [Formula: see text]. The space [Formula: see text] is called the pomset block space and we determine the complete weight distribution of it. Further, [Formula: see text]-perfect pomset block codes for ideals with partial and full counts are described. Then, for block codes with chain pomset, the packing radius and Singleton bound are established. The relation between maximum distance separable (MDS) codes and [Formula: see text]-perfect codes for any ideal [Formula: see text] is investigated. Moreover, the duality theorem for an MDS pomset block code is established when all the blocks have the same size.

Local groups in Delone sets in the Euclidean space.

A Delone (Delaunay) set is a uniformly discrete and relatively dense set of points located in space, and is a natural mathematical model of the set of atomic positions of any solid, whether it is crystalline, quasi-crystalline or amorphous. A Delone set has two positive parameters: r is the packing radius and R is the covering radius. The value 2r can be interpreted as the minimum distance between points of the set. The covering radius R is the radius of the biggest `empty' ball, i.e. the radius of the biggest ball containing no points from the set. The central concept of this article is the so-called local group at a point of X which is defined as a group of a cluster (neighborhoods) around the point of radius 2R. This value 2R is notable because it is the minimum size of cluster that provides the finiteness of the cluster group at each point for any set X from the family of all Delone sets with the covering radius R. A few conjectures and theorems on the local groups for arbitrary Delone sets in the Euclidean plane and 3D space are discussed. Some of these statements significantly refine and generalize the famous Bravais theorem on the impossibility of fifth-order axes in 2D and 3D lattices. A complete proof is given that, in a Delone set X in the 3D Euclidean space, the subset \widetilde{X} of all points at which the local groups contain rotations of order at most 6 is also a Delone set with a certain covering radius \widetilde{R}, where \widetilde{R} < 3R and R is the covering radius for X.

Sand Blocking Mechanisms and Productivity Analysis of Slant Well in Siltstone Reservoirs

The siltstone reservoir with many small layers of pay zones is usually produced by slant holes. However, severe sand blockages take place when the siltstone reservoir is developed by slant holes. Currently, the sand blocking mechanisms and the effect of sand blocking on productivity of slant well are still challenges for engineers and scholars. In this paper, based on the existing productivity model of inclined slant wells, the mechanical skin factor, which describes the effect of sand blocking on productivity, is proposed. Meanwhile, many experimental works have been done to investigate the sand blocking mechanisms in siltstone reservoirs. From the experimental work, it is concluded that with the increase of displacement PV number and displacement flow rate, the permeability of sand control system decreases by 40%. When solid particles enter the casing and block the gravel and sand control screen in the annulus, the skin factor increases sharply and the productivity decreases by more than 80%. Through the productivity calculation of multilayer sand control wells, it is considered that larger gravel packing radius can keep particles away from the well bore, which is helpful to ensure oil well productivity. Furthermore, the influence of differential filling radius on the fluid production capacity of each layer of sand control well is analyzed. It is proved that optimizing the filling radius of each layer can improve the production effect of mediate- and low-permeability layers. This method has been applied in Kendong #12 block. The daily oil production rate is increased by 9.61 t/day, and the oil recovery of this block is increased by 2.12%.

Integral trace form of extensions of degree pq

In this work, we present the integral trace form [Formula: see text] of a cyclic extension [Formula: see text] with degree [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are distinct odd primes, the conductor of [Formula: see text] is a square free integer, and [Formula: see text] belongs to the ring of algebraic integers [Formula: see text] of [Formula: see text]. The integral trace form of [Formula: see text] allows one to calculate the packing radius of lattices constructed via the canonical (or twisted) homomorphism of submodules of [Formula: see text].

Constructions of Dense Lattices of Full Diversity

A lattice construction using Z-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number field can be calculated by the trace form of the field restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subfield of a cyclotomic field. Our focus is on totally real number fields since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

Constructions of Dense Lattices of Full Diversity

A lattice construction using Z-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number field can be calculated by the trace form of the field restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subfield of a cyclotomic field. Our focus is on totally real number fields since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

Mass transfer area in a multi-stage high-speed disperser with split packing

Abstract In this study, the mean droplet diameter in the cavity zone and the total mass transfer area of a multi-stage high-speed disperser (HSD) reactor with different packing combinations were measured and evaluated. The effects of rotational speed and packing radius, as well as the packing ring radius and numbers, on the mean droplet diameter and the total mass transfer area were evaluated. A model was established to calculate the mass transfer area in the cavity zone in the HSD reactor, and it was found that the packings contribute 61%–82% of the total mass transfer area. A correlation for predicting the mass transfer area in the packing zone was regressed by the dimensionless analysis method. An enhancement factor based on the mass transfer area in the packing zone was proposed to evaluate the effect of packing combination on mass transfer area. Two optimum packing combinations were proposed in consideration of the mean droplet diameter and the enhancement factor.

Metrics Based on Finite Directed Graphs and Coding Invariants

Given a finite directed graph with $n$ vertices, we define a metric $d_{G}$ on $\mathbb {F}_{q}^{n}$ , where $\mathbb {F}_{q}$ is the finite field with $q$ elements. The weight of a word is defined as the number of vertices that can be reached by a directed path from a vertex within the support of the vector. Two canonical forms, which do not affect the metric, are given to each graph. Based on these forms we characterize each such metric. We further use these forms to prove that two graphs with different canonical forms yield different metrics. Efficient algorithms to check if a set of metric weights define a metric based on a graph are given. We provide tight bounds on the number of metric weights required to reconstruct the metric. Furthermore, we give a complete description of the group of linear isometries of the graph metrics and a characterization of the graphs for which every linear code admits a $G$ -canonical decomposition. Considering those graphs, we are able to derive an expression of the packing radius of linear codes in such metric spaces. Finally, given a directed graph which determines a hierarchical poset, we present sufficient and necessary conditions to ensure the validity of the MacWilliams identity and the MacWilliams extension property.

Generalized packing radius theorems of Alexandrov spaces with curvature ≥ 1

In this paper, we give some generalized packing radius theorems of an [Formula: see text]-dimensional Alexandrov space [Formula: see text] with curvature [Formula: see text]. Let [Formula: see text] be any [Formula: see text]-separated subset in [Formula: see text] (i.e. the distance [Formula: see text] for any [Formula: see text]). Under the condition “[Formula: see text]” (after [K. Grove and F. Wilhelm, Hard and soft packing radius theorems, Ann. of Math. 142 (1995) 213–237]), we give the upper bound of [Formula: see text] (which depends only on [Formula: see text]), and classify the geometric structure of [Formula: see text] when [Formula: see text] attains the upper bound. As a corollary, we get an isometrical sphere theorem in Riemannian case.

Liquid flow pattern transition, droplet diameter and size distribution in the cavity zone of a rotating packed bed: A visual study

A rotating packed bed (RPB) is one of the typical process intensification equipment. Its zone between the packing and the casing (cavity zone) is an important mass transfer zone. However, the liquid flow pattern, droplet diameter and size distribution in the cavity zone of a RPB, which is essential for the mass transfer modeling and performance, are still unclear. In this work, the liquid flow in the cavity zone of a RPB was studied under different experimental conditions using photographs taken by a high-speed camera. Two typical liquid flow patterns and liquid disintegrating modes were observed. A flow pattern transition criterion for the transition of ligament flow to droplet flow was presented. Effects of rotational speed, liquid initial velocity, outer packing radius, liquid viscosity, and liquid surface tension on the average droplet diameter and droplet size distribution were investigated. Furthermore, a correlation of the average droplet diameter was proposed. The predicted values of the average droplet diameter were found to be in agreement with the experimental values with deviations generally within ±15%.

Perfect codes in the [formula omitted] metric

We investigate perfect codes in Zn in the ℓp metric. Upper bounds for the packing radius r of a linear perfect code, in terms of the metric parameter p and the dimension n are derived. For p=2 and n=2,3, we determine all radii for which there exist linear perfect codes. The non-existence results for codes in Zn presented here imply non-existence results for codes over finite alphabets Zq, when the alphabet size is large enough, and have implications on some recent constructions of spherical codes.

The packing radius of a poset block code

We investigate the properties of the packing radius of a code with respect to poset block metric. In the process, we have addressed a few minor errors in the paper, “The packing radius of a code and partitioning problems: The case for poset metrics”, in Proc. IEEE Int. Symp. Information Theory (2014), pp. 2954–2958 by D’Oliveira and Firer.

Experimental study and mathematical modeling of breakthrough curve in rotating packed bed

Continuous adsorption of methylene blue (MB) from the aqueous solution onto raw rice husk was studied in rotating packed bed (RPB). A mathematical model developed for analyzing the data gave fairly good representation of the experimental breakthrough curves. Comparison with experimental data available in the literature for continuous operation in fixed bed showed that the uptake rate was an order higher in RPB. The simulation study indicated among others, that the global mass transfer coefficient decreases with increase of the radial distance from the inner packing radius, and the performance of a series of interconnected smaller rotating packed beds is superior in comparison to that of a single large bed having the same total volume and amount of adsorbent as the smaller ones.

The packing radius of a code and partitioning problems: The case for poset metrics on finite vector spaces

Until this work, the packing radius of a poset code was only known in the cases where the poset was a chain, a hierarchy, a union of disjoint chains of the same size, and for some families of codes. Our objective is to approach the general case of any poset. To do this, we will divide the problem into two parts.The first part consists in finding the packing radius of a single vector. We will show that this is equivalent to a generalization of a well known NP-hard problem known as “the partition problem”. Then, we will review the main results known about this problem giving special attention to the algorithms to solve it. The main ingredient to these algorithms is what is known as the differentiating method, which we will extend to the general case.The second part consists in finding the vector that determines the packing radius of the code. For this, we will show how it is sometimes possible to compare the packing radius of two vectors without calculating them explicitly.

PARTIAL COVERING OF A CIRCLE BY EQUAL CIRCLES. PART II: THE CASE OF 5 CIRCLES

How must n equal circles of given radius r be placed so that they cover as great a part of the area of the unit circle as possible? In this Part II of a two-part paper, a conjectured solution of this problem for n = 5 is given for r varying from the maximum packing radius to the minimum covering radius. Results are obtained by applying a mechanical model described in Part I. A generalized tensegrity structure is associated with a maximum area configuration of the 5 circles, and by using catastrophe theory, it is pointed out that the ''equilibrium paths'' have bifurcations, that is, at certain values of r, the type of the tensegrity structure and so the type of the circle configuration changes.

Practical Encoders and Decoders for Euclidean Codes from Barnes-Wall Lattices

We address the application of Barnes-Wall (BW) lattice codes for communication over additive white Gaussian noise (AWGN) channels. We introduce Construction A^prime of complex BW lattices that makes new connection between linear codes over polynomial rings and lattices. We show that Construction A^prime of BW lattices is equivalent to the multilevel construction from Reed-Muller codes proposed by Forney. To decode the BW lattice code, we adapt the low-complexity sequential BW lattice decoder (SBWD) proposed by Micciancio and Nicolosi. First we study the error performance of SBWD for decoding the infinite lattice, and demonstrate that it is powerful in making correct decisions well beyond the packing radius. Subsequently, we use the SBWD to decode lattice codes through a novel noise trimming technique, where the received vector is appropriately scaled before applying the SBWD. We show that the noise trimming technique is most effective for decoding BW lattice codes in smaller dimensions, while the gain diminishes for decoding codes in larger dimensions.

The Laplacian lattice of a graph under a simplicial distance function

We provide a complete description of important geometric invariants of the Laplacian lattice of a multigraph under the distance function induced by a regular simplex, namely Voronoi Diagram, Delaunay Triangulation, Delaunay Polytope and its combinatorial structure, Shortest Vectors, Covering Radius and Packing Radius. We use this information to obtain the following results: i. We provide an explicit description of the vertices, edges and facets of the Delaunay polytope of the origin under the simplicial distance function and using the description, we show that the Delaunay triangulation contains complete information of the multigraph up to isomorphism. ii. A weak generalization of the matrix-tree theorem: the number of multigraphs with a given Laplacian lattice is controlled, in particular upper bounded, by the number of different Delaunay triangulations. iii. We obtain formulae for the covering and packing densities of a Laplacian lattice and deduce that in the space of Laplacian lattices of undirected connected multigraphs, the Laplacian lattices of highly connected multigraphs such as Ramanujan multigraphs possess high covering and packing properties in the space of Laplacian lattices of connected graphs.

Permutation codes with specified packing radius

Most papers on permutation codes have concentrated on the minimum Hamming distance of the code. An (n, d) permutation code (or permutation array) is simply a set of permutations on n elements in which the Hamming distance between any pair of distinct permutations (or codewords) is at least d. An (n, 2e + 1) or (n, 2e + 2) permutation code is able to correct up to e errors. These codes have a potential application to powerline communications. It is known that in an (n, 2e) permutation code the balls of radius e surrounding the codewords may all be pairwise disjoint, but usually some overlap. Thus an (n, 2e) permutation code is generally unable to correct e errors using nearest neighbour decoding. On the other hand, if the packing radius of the code is defined as the largest value of e for which the balls of radius e are all pairwise disjoint, a permutation code with packing radius e can be denoted by [n, e]. Such a permutation code can always correct e errors. In this paper it is shown that, in almost all cases considered, the number of codewords in the best [n, e] code found is substantially greater than the largest number of codewords in the best known (n, 2e + 1) code. Thus the packing radius more accurately specifies the requirement for an e-error-correcting permutation code than does the minimum Hamming distance. The techniques used include construction by automorphism group and several variations of clique search They are enhanced by two theoretical results which make the computations feasible.

Canonical- systematic form for codes in hierarchical poset metrics

In this work we present a canonical-systematic form of a generator matrix for linear codes whith respect to a hierarchical poset metric on the linear space $\mathbb F_q^n$. We show that up to a linear isometry any such code is equivalent to the direct sum of codes with smaller dimensions. The canonical-systematic form enables to exhibit simple expressions for the generalized minimal weights (in the sense defined by Wei), the packing radius of the code, characterization of perfect codes and also syndrome decoding algorithm that has (in general) exponential gain when compared to usual syndrome decoding.

Classification of Niederreiter–Rosenbloom–Tsfasman Block Codes

Poset and block metrics were introduced in recent years as alternative metrics to study error correcting codes. Poset-block codes were introduced in 2008, encompassing both poset and block metrics. In this paper, we study a family of such metrics, the Niederreiter-Rosenbloom-Tsfasman block metrics. In this context, we classify the classes of equivalent codes, describe canonical representatives of each class and develop much of the classical theory of error correcting codes for Niederreiter-Rosenbloom-Tsfasman block codes, including determination of packing radius and classification of MDS and perfect codes, determination of covering radius and characterization of quasi-perfect codes, and the description of an algorithm for syndrome decoding.