Complete Sets of Mutually Orthogonal Hypercubes and Their Connections to Affine Resolvable Designs

. Recently, Laywine and Mullen proved several generalizations of Bose's equivalence between the existence of complete sets of mutually orthogonal Latin squares of order n and the existence of affine planes of order n. Laywine further investigated the relationship between sets of orthogonal frequency squares and affine resolvable balanced incomplete block designs. In this paper we generalize several of Laywine's results that were derived for frequency squares. We provide sufficient conditions for construction of an affine resolvable design from a complete set of mutually orthogonal Youden frequency hypercubes; we also show that, starting with a complete set of mutually equiorthogonal frequency hypercubes, an analogous construction can always be done. In addition, we give conditions under which an affine resolvable design can be converted to a complete set of mutually orthogonal Youden frequency hypercubes or a complete set of mutually equiorthogonal frequency hypercubes.     

Properties of complete sets of mutually equiorthogonal frequency hypercubes

A complete set of mutually equiorthogonal frequency hypercubes (MEFH) of ordern and dimensiond, usingm distinct symbols, has (n−1) ^d /(m−1) hypercubes. In this article, we explore the properties of complete sets of MEFH. As a consequence of these properties, we show that existence of such a set implies that the number of symbolsm is a prime power. We also establish an equivalence between existence of a complete set of MEFH and existence of a certain complete set of Latin hypercubes and a certain complete orthogonal array.     

Equiorthogonal Frequency Hypercubes: Preliminary Theory

Using a strong definition of frequency hypercube, we define a strengthened form of orthogonality, called equiorthogonality, for sets of such hypercubes. We prove that the maximum possible number of mutually equiorthogonal frequency hypercubes (MEFH) of order n and dimension d based on m distinct symbols is (n-1)^d/(m-1). A set of (n-1)^d/(m-1) such MEFH is called a complete set. Because of the stronger conditions on the hypercubes, we can find complete sets of MEFH of all lower dimensions within any complete set of MEFH; this useful property is not shared by sets of mutually orthogonal hypercubes using the usual, weaker definition.     

Affine geometries and sets of equiorthogonal frequency hypercubes

Equiorthogonal frequency hypercubes are one particular generalization of orthogonal latin squares. A complete set of mutually equiorthogonal frequency hypercubes (MEFH) of order n and dimension d, using m distinct symbols, has (n − 1)^d/(m − 1) hypercubes. In this article, we prove that an affine geometry of dimension dh over 𝔽_m can always be used to construct a complete set of MEFH of order m^h and dimension d, using m distinct symbols. We also provide necessary and sufficient conditions for a complete set of MEFH to be equivalent to an affine geometry. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 435–441, 2000     

An affine design with v = m2h and k = m2h−1 not equivalent to a complete set of F(mh; mh−1) MOFS

Existing sufficient conditions for the construction of a complete set of mutually orthogonal frequency squares from an affine resolvable design are improved to give necessary and sufficient conditions. In doing so a design is exhibited that proves that the class of complete sets of MOFS under consideration is a proper subset of the class of affine resolvable designs with matching parameters. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 331–340, 1999     

D-Dimensional hypercubes and the Euler and MacNeish conjectures

We give a construction for large sets of mutually orthogonal hypercubes of dimensionald given sets of mutually orthogonal latin squares and hypercubes of lower dimension. We also considerd>-2 dimensional versions of the Euler and MacNeish conjectures as well as discussing applications to improved constructions of (t, m, s)-nets, useful in pseudorandom number generation and quasi-Monte-Carlo methods of numerical integration.This author would like to thank the National Security Agency for partial support under grant agreement # MDA 904-92-H-3044.     

Point sets with uniformity properties and orthogonal hypercubes

The theory of (t, m, s)-nets is useful in the study of sets of points in the unit cube with small discrepancy. It is known that the existence of a (0, 2,s)-net in baseb is equivalent to the existence ofs–2 mutually orthogonal latin squares of orderb. In this paper we generalize this equivalence by showing that fort0 the existence of a (t, t+2,s)-net in baseb is equivalent to the existence ofs mutually orthogonal hypercubes of dimensiont+2 and orderb. Using the theory of hypercubes we obtain upper bounds ons for the existence of such nets. Forb a prime power these bounds are best possible. We also state several open problems.This author would like to thank the Mathematics Department of the University of Tasmania for its hospitality during his sabbatical when this paper was written. The same author would also like to thank the NSA for partial support under grant agreement # MDA904-87-H-2023.This author's research was supported by a grant from the Commonwealth of Australia through the Australian Research Council.     

A note on orthogonal sets of hypercubes

Mutually orthogonal sets of hypercubes are higher dimensional generalizations of mutually orthogonal sets of Latin squares. For Latin squares, it is well known that the Cayley table of a group of order n is a Latin square, which has no orthogonal mate if n is congruent to 2 modulo 4. We will prove an analogous result for hypercubes. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 231–233, 1997     

A combinatorial theory of Grünbaum's new regular polyhedra,Part II: Complete enumeration

The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE ³ and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE ³, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE ⁿ is worked out in §9.     

The geometry of sets of orthogonal frequency hypercubes

We extend the notion of a framed net, introduced by D. Jungnickel, V. C. Mavron, and T. P. McDonough, J Combinatorial Theory A, 96 (2001), 376–387, to that of a d‐framed net of type ?, where d ≥ 2 and 1 ≤ ? ≤ d‐1, and we establish a correspondence between d‐framed nets of type ? and sets of mutually orthogonal frequency hypercubes of dimension d. We provide a new proof of the maximal size of a set of mutually orthogonal frequency hypercubes of type ? and dimension d, originally established by C. F. Laywine, G. L. Mullen, and G. Whittle, Monatsh Math 119 (1995), 223–238, and we obtain a geometric characterization of the framed net when this bound is satisfied as a PBIBD based on a d‐class association Hamming scheme H(d,n). © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 449–459, 2007     

Sets of orthogonal hypercubes of class r

A (d,n,r,t)-hypercube is an n×n×?×n (d-times) array on nr symbols such that when fixing t coordinates of the hypercube (and running across the remaining d−t coordinates) each symbol is repeated nd−r−t times. We introduce a new parameter, r, representing the class of the hypercube. When r=1, this provides the usual definition of a hypercube and when d=2 and r=t=1 these hypercubes are Latin squares. If d?2r, then the notion of orthogonality is also inherited from the usual definition of hypercubes. This work deals with constructions of class r hypercubes and presents bounds on the number of mutually orthogonal class r hypercubes. We also give constructions of sets of mutually orthogonal hypercubes when n is a prime power.     

Eigenschaften von vollstaendigen Systemen orthogonaler lateinischer Quadrate,die bestimmte affine Ebenen repraesentieren

A set of n-1 mutually orthogonal Latin squares of order n is a model of an affine plane with exactly n points on a line and every affine plane with n points on a line can be represented by n-1 mutually orthogonal Latin squares ([1]). In this paper we investigate properties of finite planes through the complete set of mutually orthogonal Latin squares representing the plane and mainly — vice versa — properties of the squares representing a fixed plane. The results are based on the geometrical configurations which hold in the planes. For presumed definitions and theorems which are not specially referred to see [4], [7], [3] or [6].     

Equivalence classes of central semiregular relative difference sets

A semiregular relative difference set (RDS) in a finite group E which avoids a central subgroup C is equivalent to a cocycle which satisfies an additional condition, called orthogonality. However the basic equivalence relation, cohomology, on cocycles, does not preserve orthogonality, leading to the perception that orthogonality is essentially a combinatorial property. We show this perception is false by discovering a natural atomic structure within cohomology classes, which discriminates between orthogonal and non‐orthogonal cocycles. This atomic structure is determined by an action we term the shift action of the group G = E/C on cocycles, which defines a stronger equivalence relation on cocycles than cohomology. We prove that for each triple (C, E, G), the set of equivalence classes of semiregular RDS in E relative to C is in one to one correspondence with the set of shift‐orbits of the (Aut(C) × Aut(G))‐orbits of orthogonal cocycles. This determines a new algorithm for detecting and classifying central semiregular RDS. We demonstrate it, and propose a 7‐parameter classification scheme for equivalence classes of central semiregular relative difference sets. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 330–346, 2000     

Affine Mendelsohn triple systems and the Eisenstein integers

We define a Mendelsohn triple system (MTS) of order coprime with 3, and having multiplication affine over an abelian group, to be affine, nonramified. By exhibiting a one‐to‐one correspondence between isomorphism classes of affine MTS and those of modules over the Eisenstein integers, we solve the isomorphism problem for affine, nonramified MTS and enumerate these isomorphism classes (extending the work of Donovan, Griggs, McCourt, Opr?al, and Stanovský). As a consequence, all entropic MTSs of order coprime with 3 and distributive MTS of order coprime with 3 are classified. Partial results on the isomorphism problem for affine MTS with order divisible by 3 are given, and a complete classification is conjectured. We also prove that for any affine MTS, the qualities of being nonramified, pure, and self‐orthogonal are equivalent.     

Rigid pentagons in hypercubes

Two sets of vertices of a hypercubes in ⁿ and ^m are said to be equivalent if there exists a distance preserving linear transformation of one hypercube into the other taking one set to the other. A set of vertices of a hypercube is said to be weakly rigid if up to equivalence it is a unique realization of its distance pattern and it is called rigid if the same holds for any multiple of its distance pattern. A method of describing all rigid and weakly rigid sets of vertices of hypercube of a given size is developed. It is also shown that distance pattern of any rigid set is on the face of convex cone of all distance patterns of sets of vertices in hypercubes.Rigid pentagons (i.e. rigid sets of size 5 in hypercubes) are described. It is shown that there are exactly seven distinct types of rigid pentagons and one type of rigid quadrangle. It is also shown that there is a unique weakly rigid pentagon which is not rigid. An application to the study of all rigid pentagons and quadrangles inL ¹ having integral distance pattern is also given.This work was done during a visit of both the authors to Mehta Research Institute, Allahabad, India.     

Subsquares in orthogonal latin squares as subspaces in affine geometries: A generalization of an equivalence of bose

The combinatorial properties of subsquares in orthogonal latin squares are examined. Using these properties it is shown that in appropriate orthogonal latin squares of orderm ^h blocks of subsquares of orderm ^h(i–1)/i , wherei dividesh, form the hyperplanes of the affine geometryAG (2i, m ^h/i ). This means that a given set of mutually orthogonal latin squares may be equivalent simultaneously to a number of different geometries depending on the order of the subsquares used to form the hyperplanes. In the case thati=1, the subsquares become points, the hyperplanes become lines, and the equivalence reduces to the well known result of Bose relating orthogonal latin squares and affine planes.The author would like to thank the Natural Sciences and Engineering Research Council of Canada for partial support under grant no. OGP0014645.     

Sudoku-like arrays,codes and orthogonality

This paper is concerned with constructions and orthogonality of generalized Sudoku arrays of various forms. We characterize these arrays based on their constraints; for example Sudoku squares are characterized by having strip and sub-square constraints. First, we generalize Sudoku squares to be multi-dimensional arrays with strip and sub-cube constraints and construct mutually orthogonal sets of these arrays using linear polynomials. We add additional constraints motivated by elementary intervals for low discrepancy sequences and again give a construction of these arrays using linear polynomials in detail for 3 dimensional and a general construction method for arbitrary dimension. Then we give a different construction of these hypercubes due to MDS codes. We also analyze the orthogonality of all of the Sudoku-like hypercubes we consider in this paper.     

Relative difference sets,graphs and inequivalence of functions between groups

For cryptographic purposes, we want to find functions with both low differential uniformity and dissimilarity to all linear functions and to know when such functions are essentially different. For vectorial Boolean functions, extended affine equivalence and the coarser Carlet–Charpin–Zinoviev (CCZ) equivalence are both used to distinguish between nonlinear functions. It remains hard to tell when CCZ equivalent functions are EA‐inequivalent. This paper presents a framework for solving this problem in full generality, for functions between arbitrary finite groups. This common framework is based on relative difference sets (RDSs). The CCZ and EA equivalence classes of perfect nonlinear (PN) functions are each derived, by quite different processes, from equivalence classes of splitting semiregular RDSs. By generalizing these processes, we obtain a much strengthened formula for all the graph equivalences which define the EA equivalence class of a given function, amongst those which define its CCZ equivalence class. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 260–273, 2010     

On orthogonal orthomorphisms of cyclic and non‐abelian groups. II

We construct sets of three pairwise orthogonal orthomorphisms of Z_3n, n not divisible by either 2 or 3, n ≠ 7, 17. Combined with results in the literature, this reduces the problem of determining for which v, there exist three pairwise orthogonal orthomorphisms of Z_v to the case v = 9p, p > 3 a prime. This yields new lower bounds for the number of pairwise orthogonal orthomorphisms of classes of dihedral groups of doubly even order, and classes of linear groups. These results also find application in the construction of Z‐cyclic triplewhist tournaments. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 195–209, 2007     

A Note on the Construction of Orthogonal Latin Hypercube Designs

Latin hypercube designs have been found very useful for designing computer experiments. In recent years, several methods of constructing orthogonal Latin hypercube designs have been proposed in the literature. In this article, we report some more results on the construction of orthogonal Latin hypercubes which result in several new designs.