On symmetric nets and generalized Hadamard matrices from affine designs

Symmetric nets are affine resolvable designs whose duals are also affine. It is shown that. up to isomorphism, there are exactly four symmetric (3, 3)-nets (v=b=27,k=9), and exactly two inequivalent 9×9 generalized Hadamard matrices over the group of order 3. The symmetric (3, 3)-nets are found as subnets of affine resolvable 2-(27, 9, 4) designs. Ten of the 68 non-isomorphic affine resolvable 2-(27, 9, 4) designs are not extensions of symmetric (3, 3)-subnets, providing the first examples of affine 2-(q³, q², q²–1/q–1) designs without symmetric (q, q)-subnets.     

A technique for constructing non‐embeddable quasi‐residual designs

We propose a technique for constructing two infinite families of non‐embeddable quasi‐residual designs as soon as one such design satisfying certain conditions exists. The main tools are generalized Hadamard matrices and balanced generalized weighing matrices. Starting with a specific non‐embeddable quasi‐residual 2‐(27,9,4) design, we construct for every positive integer m a non‐embeddable 2‐(3^m,3^m?1,(3^m?1?1)/2)‐design, and, if r_m=(3^m?1)/2 is a prime power, we construct for every positive integer n a non‐embeddable design. For each design in these families, a symmetric design with the corresponding parameters is known to exist. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 160–172, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.900     

Building Symmetric Designs With Building Sets

We introduce a uniform technique for constructing a family of symmetric designs with parameters (v(q ^m+1-1)/(q-1), kq ^m ,q ^m), where m is any positive integer, (v, k, ) are parameters of an abelian difference set, and q = k ²/(k - ) is a prime power. We utilize the Davis and Jedwab approach to constructing difference sets to show that our construction works whenever (v, k, ) are parameters of a McFarland difference set or its complement, a Spence difference set or its complement, a Davis–Jedwab difference set or its complement, or a Hadamard difference set of order 9 · 4 ^d , thus obtaining seven infinite families of symmetric designs.     

Primitive symmetric designs with up to 2500 points

By this article we conclude the construction of all primitive ( v, k,λ ) symmetric designs with v < 2500 , up to a few unsolved cases. Complementary to the designs with prime power number of points published previously, here we give 55 primitive symmetric designs with v ≠ p ^m , p prime and m positive integer, together with the analysis of their full automorphism groups. The research involves programming and wide‐range computations. We make use of the software package GAP and the library of primitive groups which it contains. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:463‐474, 2011     

The spectrum of additive BIB designs

In this article, the existence of additive BIB designs is discussed with direct and recursive constructions, together with investigation of a property of resolvability. Such designs can be used to construct infinite families of BIB designs. In particular, we obtain a series of B(sⁿ, ts^m, λt (ts^m ? 1) (s^n‐m ? 1)/[2(s^m ? 1)]) for any positive integer λ, such that sⁿ (sⁿ ? 1) λ ≡ 0 (mod s^m (s^m ? 1) and for any positive integer t with 2 ≤ t ≤ s^n‐m, where s is an odd prime power. Connections between additive BIB designs and other combinatorial objects such as multiply nested designs and perpendicular arrays are discussed. A construction of resolvable BIB designs with v = 4k is also proposed. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 235–254, 2007     

Quasi‐symmetric designs with fixed difference of block intersection numbers

The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.

• (1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z³ then λ < x + 1 + z + z³.
• (2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
• (3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z².
• (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
• (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.

© 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007     

Alias balanced and alias partially balanced fractional 2 m factorial designs of resolution 2l+1

As a generalization of alias balanced designs due to Hedayat, Raktoe and Federer [5], we introduce the concept of alias partially balanced designs for fractional 2 ^m factorial designs of resolution 2l+1. All orthogonal arrays of strength 2l yield alias balanced designs. Some balanced arrays of strength 2l yield alias balanced and alias partially balanced designs. In particular, simple arrays which are a special case of balanced arrays yield alias partially balanced designs. At most 2 ^m −1 alias balanced (or alias partially balanced) designs are generated from an alias balanced (or alias partially balanced) design by level permutations. This implies that alias balanced or alias partially balanced designs need not be orthogonal arrays or balanced arrays of strength 2l.     

A family of non class-regular symmetric transversal designs of spread type

Many classes of symmetric transversal designs have been constructed from generalized Hadamard matrices and they are necessarily class regular. In (Hiramine, Des Codes Cryptogr 56:21–33, 2010) we constructed symmetric transversal designs using spreads of \mathbbZ_p²ⁿ{\mathbb{Z}_p^{2n}} with p a prime. In this article we show that most of them admit no class regular automorphism groups. This implies that they are never obtained from generalized Hadamard matrices. As far as we know, this is the first infinite family of non class-regular symmetric transversal designs.     

A Recursive Construction for New Symmetric Designs

We introduce a recursive construction of regular Handamard matrices with row sum 2h for h=±3ⁿ. Whenever q=(2h – 1)² is a prime power, we construct, for every positive integer m, a symmetric designs with parameters (4h²(q^m+1 – 1)/(q – 1), (2h² – h)q^m, (h² – h)q^m).     

Decomposing complete equipartite graphs into odd square‐length cycles: number of parts odd

In this article, we introduce a new technique for obtaining cycle decompositions of complete equipartite graphs from cycle decompositions of related multigraphs. We use this technique to prove that if n, m and λ are positive integers with n ≥ 3, λ≥ 3 and n and λ both odd, then the complete equipartite graph having n parts of size m admits a decomposition into cycles of length λ² whenever nm ≥ λ² and λ divides m. As a corollary, we obtain necessary and sufficient conditions for the decomposition of any complete equipartite graph into cycles of length p², where p is prime. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:401‐414, 2010     

Totally geodesic submanifolds of the complex quadric

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Qm:=SO(m+2)/(SO(2)×SO(m)) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Qm by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Qm are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]: The first type is constituted by manifolds isometric to CP¹×RP¹; their existence follows from the fact that Q² is (via the Segre embedding) holomorphically isometric to CP¹×CP¹. The second type consists of 2-spheres of radius which are neither complex nor totally real in Qm.     

Symmetric designs with bruck subdesigns

IfP is a finite projective plane of ordern with a proper subplaneQ of orderm which is not a Baer subplane, then a theorem of Bruck [Trans. AMS 78(1955), 464–481] asserts thatn≧m ²+m. If the equalityn=m ²+m were to occur thenP would be of composite order andQ should be called a Bruck subplane. It can be shown that if a projective planeP contains a Bruck subplaneQ, then in factP contains a designQ′ which has the parameters of the lines in a three dimensional projective geometry of orderm. A well known scheme of Bruck suggests using such aQ′ to constructP. Bruck’s theorem readily extends to symmetric designs [Kantor, Trans. AMS 146 (1969), 1–28], hence the concept of a Bruck subdesign. This paper develops the analoque ofQ′ and shows (by example) that the analogous construction scheme can be used to find symmetric designs.     

2-Designs overGF(q)

We give a construction of a series of 2-(n, 3,q ²+q+1;q) designs of vector spaces over a finite fieldGF(q) of odd characteristic. These designs correspond to those constructed by Thomas and the author for even characteristic. As a natural generalization we give a collection ofm-dimensional subspaces which possibly become a 2-(n, m, λ; q) design.     

Symmetric Weighing Matrices Constructed using Group Matrices

A weighing matrix of order n and weight m² is a square matrix M of order n with entries from {-1,0,+1} such that MM^T=m²I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.M^T=M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2p^r and weight p² where p is a prime and p≥ 5.Communicated by: K.T. Arasu     

On the robustness of balanced fractional 2m factorial designs of resolution 2l+1 in the presence of outliers

Summary By use of the algebraic structure, we obtain a simplified expression for the outlier-insensitivity factor for balanced fractional 2^m factorial (2^m-BFF) designs of resolution 2l+1 derived from simple arrays (S-arrays), whose measure has been introduced by Ghosh and Kipnegeno (1985,J. Statist. Plann. Inference,11, 119–129). It is defined by use of the measure suggested by Box and Draper (1975,Biometrika, 62 (2), 347–352). As examples, we study the sensitivity ofA-optimal 2^m-BFF designs of resolution VII (i.e.,l=3) given by Shirakura (1976,Ann. Statist.,4, 515–531; 1977,Hiroshima Math. J.,7, 217–285). We observe that these designs are robust in the sense that they have low sensitivities. Research supported in part by Grant 59530012 (C) and 60530014 (C), Japan.     

A Class of 2‐(3n7, 3n−17, (3n−17−1)/2) Designs

Generalized Hadamard matrices are used for the construction of a class of quasi‐residual nonresolvable BIBD's with parameters . The designs are not embeddable as residual designs into symmetric designs if n is even. The construction yields many nonisomorphic designs for every given n ≥ 2, including more than 10¹⁷ nonisomorphic 2‐(63,21,10) designs. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 460–464, 2007     

Multiple Symmetric Solutions for Discrete Lidstone Boundary Value Problems

We offer criteria for the existence of single, double and multiple positive symmetric solutions for the boundary value problem ?^{2m y(k-m)= f(y(k), ?²y(k-1)….,?SUP>2i y(k-i),…,?2(m-1)} y(k-(m-1))), k∈{a+1,…,b+1} ?²ⁱ y(a+1-m)=?²ⁱ y(b+1+m-2i)=0, 0≤i≤m-1 where m ≥ 1 and (-1)^m f can either be positive or the condition can be relaxed.     

The existence of large set of symmetric partitioned incomplete latin squares

In this paper, we investigate the existence of large sets of symmetric partitioned incomplete latin squares of type g^u (LSSPILSs) which can be viewed as a generalization of the well‐known golf designs. Constructions for LSSPILSs are presented from some other large sets, such as golf designs, large sets of group divisible designs, and large sets of Room frames. We prove that there exists an LSSPILS(g^u) if and only if u ≥ 3, g(u ? 1) ≡ 0 (mod 2), and (g, u) ≠ (1, 5).     

Exact analysis of asymmetric random polling systems with single buffers and correlated input process

We introduce a simple approach for modeling and analyzing asymmetric random polling systems with single buffers and correlated input process. We consider two variations of single buffers system: the conventional system and the buffer relaxation system. In the conventional system, at most one customer may be resided in any queue at any time. In the buffer relaxation system, a buffer becomes available to new customers as soon as the current customer is being served. Previous studies concentrate on conventional single buffer system with independent Poisson process input process. It has been shown that the asymmetric system requires the solution ofm 2 ^m –1) linear equations; and the symmetric system requires the solution of 2 ^m–1–1 linear equations, wherem is the number of stations in the system. For both the conventional system and the buffer relaxation system, we give the exact solution to the more general case and show that our analysis requires the solution of 2 ^m –1 linear equations. For the symmetric case, we obtain explicit expressions for several performance measures of the system. These performance measures include the mean and second moment of the cycle time, loss probability, throughput, and the expected delay observed by a customer.     

The retraction algorithm for factoring banded symmetric matrices

Let A be an n × n symmetric matrix of bandwidth 2m + 1. The matrix need not be positive definite. In this paper we will present an algorithm for factoring A which preserves symmetry and the band structure and limits element growth in the factorization. With this factorization one may solve a linear system with A as the coefficient matrix and determine the inertia of A, the number of positive, negative, and zero eigenvalues of A. The algorithm requires between 1/2nm² and 5/4nm² multiplications and at most (2m + 1)n locations compared to non‐symmetric Gaussian elimination which requires between nm² and 2nm² multiplications and at most (3m + 1)n locations. Our algorithm reduces A to block diagonal form with 1 × 1 and 2 × 2 blocks on the diagonal. When pivoting for stability and subsequent transformations produce non‐zero elements outside the original band, column/row transformations are used to retract the bandwidth. To decrease the operation count and the necessary storage, we use the fact that the correction outside the band is rank‐1 and invert the process, applying the transformations that would restore the bandwidth first, followed by a modified correction. This paper contains an element growth analysis and a computational comparison with LAPACKs non‐symmetric band routines and the Snap‐back code of Irony and Toledo. Copyright © 2007 John Wiley & Sons, Ltd.