Orthogonal Partitions Research Articles

Construction of Optimal Orthogonal Partition

ABSTRACTOrthogonal partitions play a crucial role in orthogonal array theory, design of experiments and quantum information theory. The optimisation of orthogonal partitions can improve the saturation percentages of orthogonal arrays (OAs) obtained by the orthogonal partition method. In particular, optimal orthogonal partitions of strength 1 are of great practical utility. However, there is still a scarcity of results about orthogonal partitions, especially optimal ones. In this paper, the definition of an optimal orthogonal partition is proposed, and we construct optimal orthogonal partitions of OAs by several construction methods, such as orthogonal partition method, difference scheme construction, generalised product construction and construction. As an application, we obtain various optimal orthogonal partitions and OAs with higher saturation percentages.

Quantum Error-Correcting Codes Based on Orthogonal Arrays.

In this paper, by using the Hamming distance, we establish a relation between quantum error-correcting codes ((N,K,d+1))s and orthogonal arrays with orthogonal partitions. Therefore, this is a generalization of the relation between quantum error-correcting codes ((N,1,d+1))s and irredundant orthogonal arrays. This relation is used for the construction of pure quantum error-correcting codes. As applications of this method, numerous infinite families of optimal quantum codes can be constructed explicitly such as ((3,s,2))s for all si≥3, ((4,s2,2))s for all si≥5, ((5,s,3))s for all si≥4, ((6,s2,3))s for all si≥5, ((7,s3,3))s for all si≥7, ((8,s2,4))s for all si≥9, ((9,s3,4))s for all si≥11, ((9,s,5))s for all si≥9, ((10,s2,5))s for all si≥11, ((11,s,6))s for all si≥11, and ((12,s2,6))s for all si≥13, where s=s1⋯sn and s1,…,sn are all prime powers. The advantages of our approach over existing methods lie in the facts that these results are not just existence results, but constructive results, the codes constructed are pure, and each basis state of these codes has far less terms. Moreover, the above method developed can be extended to construction of quantum error-correcting codes over mixed alphabets.

Construction of Binary Quantum Error-Correcting Codes from Orthogonal Array.

By using difference schemes, orthogonal partitions and a replacement method, some new methods to construct pure quantum error-correcting codes are provided from orthogonal arrays. As an application of these methods, we construct several infinite series of quantum error-correcting codes including some optimal ones. Compared with the existing binary quantum codes, more new codes can be constructed, which have a lower number of terms (i.e., the number of computational basis states) for each of their basis states.

Construction of mixed orthogonal arrays with high strength

A considerable portion of the work on mixed orthogonal arrays applies specifically to arrays of strength 2. Although strength t=2 is arguably the most important case for statistical applications, there is an urgent need for better methods for t≥3. However, the knowledge on the existence of arrays for t≥3 is rather limited. In this paper, new construction methods for symmetric and asymmetric orthogonal arrays (OAs) with high strength are proposed by using lower strength orthogonal partitions of spaces and OAs. A positive answer is provided to the open problem in Hedayat, Sloane and Stufken (Orthogonal Arrays: Theory and Applications (1999) Springer) on developing better methods and tools for the construction of mixed orthogonal arrays with strength t≥3. Not only are the methods straightforward, but also they are useful for constructing symmetric or asymmetric OAs of arbitrary strengths, numbers of levels and various sizes. The constructed OAs can be utilized to generate more OAs. The resulting OAs have a high degree of flexibility and many other desirable properties. Some selective OAs are tabulated for practical uses.

Quantum k-uniform states for heterogeneous systems from irredundant mixed orthogonal arrays

Quantum multipartite entangled states play significant roles in quantum information processing. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of irredundant mixed orthogonal arrays (IrMOAs) and thus provide positive answers to two open problems. The first is the extension of the method for constructing homogeneous systems from orthogonal arrays to heterogeneous multipartite systems with different individual levels. The second is the existence of k-uniform states in heterogeneous quantum systems. We present explicit constructions of two- and three-uniform states for arbitrary heterogeneous multipartite systems with coprime individual levels and characterize the entangled states in heterogeneous systems consisting of subsystems with nonprime power dimensions as well. Moreover, we obtain infinite classes of k-uniform states for heterogeneous multipartite systems for any $$k\ge 2$$ . The non-existence of a class of IrMOAs is also proved.

Isolated and structured families of models for stochastic symmetric matrices

Stochastic symmetric matrices with a dominant eigenvalue, α,can be written as the sum of λααt (where λ is the first eigenvalue), with a symmetric error matrix E. The information in the stochastic matrix will be condensed in its structured vectors, λα, and the sum of square of residues, V. When the matrices of a family correspond to the treatments of a base design, we say the family is structured. The action of the factors, which are considered in the base design, on the structure vectors of the family matrices will be analyzed. We use ANOVA (Analysis of Variance) and related techniques, to study the action under linear combinations of the components of structure vectors of the m matrices of the model. Orthogonal models with m treatments are associated to orthogonal partitions. The hypothesis to be tested, on the action of the factors in the base design, will be associated to the spaces in the orthogonal partitions. We will show how to carry out transversal and longitudinal analysis for families of stochastic symmetric matrices with dominant eigenvalue associated to orthogonal models.

The Use of Collaboration Distance in Scheduling Conference Talks

Several bibliographic databases offer a free tool that enables one to determine the collaboration distance or co-authorship distance between researchers. This paper addresses a real-life application of the collaboration distance. It concerns somewhat unusual clustering; namely clustering in which the the average distances in each cluster need to be maximised. We briefly consider a pair of clusterings in which two cluster partitions are uniform and orthogonal in the sense that in each partition all clusters are of the same size and that no pair of elements belongs to the same cluster in both partitions. We consider different objective functions when calculating the score of the pair of orthogonal partitions. The one used on the Wiener index (a graph invariant, known in chemical graph theory) is used. The main application of our work is an algorithm for scheduling a series of parallel talks in a major conference.

Two and three-uniform states from irredundant orthogonal arrays

A pure quantum state of N subsystems, each with d levels, is said to be k-uniform if all of its reductions to k qudits are maximally mixed. Only the uniform states obtained from orthogonal arrays (OAs) are considered throughout this work. The Hamming distances of OAs are specially applied to the theory of quantum information. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of irredundant orthogonal arrays (IrOAs), then answer the open questions of whether there exist 3-uniform states of N qubits and 2-uniform states of N qutrits, and whether 3-uniform states of qudits (d > 2) for high values of N can be explicitly constructed. In fact, we obtain 3-uniform states for an arbitrary number of N ≥ 8 qubits and 2-uniform states of N qutrits for every N ≥ 4. Additionally, we provide explicit constructions of the 3-uniform states of N ≥ 8 qutrits, N = 6 and N ≥ 8 ququarts and ququints, N ≥ 6 qudits having d levels for any prime power d > 6, and N = 8 and N ≥ 12 qudits having d levels for non-prime-power d ≥ 6. Moreover, we describe an explicit construction scheme for the 2-uniform states of qudits having d ≥ 4 levels. The proofs of existence of the 2-uniform states of N ≥ 6 qubits are simplified by using a class of OAs. Two special 3-uniform states are obtained from IrOA(32, 10, 2, 3) and IrOA(32, 11, 2, 3) using the interaction column property of OAs.

Iterative Constructions of Orthogonal Arrays of Strength <i>t</i> and Orthogonal Partitions

Iterative Constructions of Orthogonal Arrays of Strength <i>t</i> and Orthogonal Partitions

ALGORITHMS FOR LABELING CONSTANT WEIGHT GRAY CODES

Let n and r be positive integers with 1 < r < n, and let Xn = {1, 2 ,..., n} .A nr-set A and a partition π of Xn are said to be orthogonal if every class of π meets A in exactly one element. We prove that if A1, A2 ,..., A ( n) is a list of the distinct r-sets of Xn with |Ai ∩ Ai+1 |= r − 1f ori = 1, 2 ,..., n r taken modulo n r , then there exists a list of distinct partitions π1 ,π 2 ,...,π ( n) such that πi is orthogonal to both Ai and Ai+1. This result states that any constant weight Gray code admits a labeling by distinct orthogonal partitions. Using an algorithm from the literature on Gray codes, we provide a surprisingly efficient algorithm that on input (n, r) outputs an orthogonally labeled constant weight Gray code. We also prove a two-fold Gray enumeration result, presenting an orthogonally labeled constant weight Gray code in which the partition labels form a cycle in the covering graph of the lattice of all partitions of Xn. This leads to a conjecture related to the Middle Levels Conjecture. Finally, we provide an application of our results to calculating minimal generating sets of idempotents for finite semigroups.

Compatibility and decompositions of effects

Compatibility relations in effect algebras and their connections with refinements of the orthogonal partitions of unity are studied. Properties of blocks as maximal sets of compatible elements are discussed. Some special kinds of effect algebras are characterized using properties of compatibility. Using refinements, an additional structure on the effect test spaces is introduced and used to a characterization of different types of effect algebras.

Air-Borne Sound Insulation of Single Walls at Low Frequencies: A Discussion on the Influence of Boundary and Mounting Conditions

A theoretical analysis of the air-borne sound transmission through a single wall at low frequencies is presented. The wall is assumed to be surrounded by orthogonally mounted partitions. The results are compared with those presented previously for a wall mounted in a baffle. The comparison shows that the presence of the orthogonal partitions can substantially increase the transmission due to the forced response at frequencies around and below the critical frequency. The magnitude of this effect depends on how the wall is mounted to surrounding constructions; it can be large for certain cases. The analysis also shows that the transmission due to the resonant response increases by 6 dB at frequencies below the critical frequency in the region where the edge radiators are predominant.

Cycle reversals in oriented plane quadrangulations and orthogonal plane partitions

Aplane quadrangulation G is a simple plane graph such that each face ofG is quadrilateral. A (*) -orientation D*(G) ofG is an orientation ofG such that the outdegree of each vertex on ∂G is 1 and the outdegrees of other vertices are all 2, where ∂G denotes the outer 4-cycle ofG. In this paper, we shall show that every plane quadrangulationG has at least one (*)-orientation. We also show that any two (*)-orientations ofG can be transformed into one another by a sequence of 4-cycle reversals. Moreover, we apply this fact toorthogonal plane partitions, which are partitions of a square into rectangles by straight segments.

Orthogonal partitions in designed experiments

Orthogonal partitions in designed experiments

Orthogonal partitions in designed experiments

A survey is given of the statistical theory of orthogonal partitions on a finite set. Orthogonality, closure under suprema, and one trivial partition give an orthogonal decomposition of the corresponding vector space into subspaces indexed by the partitions. These conditions plus uniformity, closure under infima and the other trivial partition give association schemes. Examples covered by the theory include Latin squares, orthogonal arrays, semilattices of subgroups, and partitions defined by the ancestral subsets of a partially ordered set (the poset block structures). Isomorphism, equivalence and duality are discussed, and the automorphism groups given in some cases. Finally, the ideas are illustrated by some examples of real experiments.

Nonparametric rank-based test procedures for non-additive models in the two-way layout I. no replications

One of the major unresolved problems in the area of nonparametric statistics is the need for satisfactory rank-based test procedures for non-additive models in the two-way layout, especially when there is only one observation on each combination of the levels of the experimental factors. In this paper we consider an arbitrary non-additive model for the two-way layout with n levels of each factor. We utilize both alignment and ranking of the data together with basic properties of Latin squares to develop rank tests for interaction (non-additivity). Our technique involves first aligning within one of the main effects, ranking within the other main effects (columns and rows) and then adding the resulting ranks within “interaction bands” corresponding to orthogonal partitions of the interaction for the model, as denoted by the letters of an n × n Latin square. A Friedman-type statistic is then computed on the resulting sums. This is repeated for each of (n−1) mutually orthogonal Latin squares (thus accounting for all the interaction degrees of freedom). The resulting (n−1) Friedman-type statistics are finally combined to obtain an overall test statistic. The necessary null distribution tables for applying the proposed test for non-additivity are presented and we discuss the results of a Monte Carlo simulation study of the relative powers of this new procedure and other (parametric and nonparametric) procedures designed to detect interaction in a two-way layout with one observation per cell.

Orthogonal partitions and covering of graphs

Orthogonal partitions and covering of graphs

A Note on Orthogonal Partitions and Some Well-Known Structures in Design of Experiments

Finney has used orthogonal partitions in the context of the search for higher order (coarser) partitions of given Latin squares. Hedayat and Seiden use the term $F$-square to denote higher order partitions that are orthogonal to both rows and columns. This note is a short expository treatment of orthogonal partitions in general and is based on the identification of a partition with a vector subspace of Euclidean $N$-space $R^N$. This identification is not new as it is part of the usual vector space approach to analysis of variance. This approach puts the concept of orthogonal partitions in a simple light unencumbered by the language of design of experiments. Another advantage is that certain published bounds on the maximum number of orthogonal partitions of specified type are immediate from the dimensionality restriction imposed by $R^N$. In addition, some counting problems are identified which are of possible interest to researchers in design of experiments and combinatorics.

Human Factors in Man-Machine Systems

The field of human factors is analyzed in an effort to discover its scope and organization. Three orthogonal partitions of the field serve to define eight mutually exclusive, jointly exhaustive, and interdependent sectors. These are considered separately.