Edge partitions of the complete symmetric directed graph and related designs

We show that the edges of the complete symmetric directed graph onn vertices can be partitioned into directed cycles (or anti-directed cycles) of lengthn−1 so that any two distinct cycles have exactly one oppositely directed edge in common whenn=p ^e>3, wherep is a prime ande is a positive integer. When the cycles are anti-directedp must be odd. We then consider the designs which arise from these partitions and investigate their construction. We would like to thank the Mathematics Department of the University of Arizona for their hospitality during January 1979 when this work was begun. Partial support was provided for this research by the Natural Sciences and Engineering Research Council of Canada under Grant A-4792.     

On vector space partitions and uniformly resolvable designs

Let V _n (q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V _n (q) is a partition of V _n (q) if every nonzero vector in V _n (q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V _n (q) containing a _i subspaces of dimension n _i for 1 ≤ i ≤ k induces a uniformly resolvable design on q ⁿ points with a _i parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q ⁿ points where corresponding partitions of V _n (q) do not exist. A. D. Blinco—Part of this research was done while the author was visiting Illinois State University.     

Mixed block designs

Let V_i (i = 1, 2) be a set of size v_i. Let D be a collection of ordered pairs (b₁, b₂) where b_i is a k_i-element subset of V_i. We say that D is a mixed t-design if there exist constants λ _(j,j2), (0 ≤ j_i ≤ k_i, j₁ + j₂ ≤ t) such that, for every choice of a j₁-element subset S₁ of V₁ and every choice of a j₂-element subset S₂ of V₂, there exist exactly λ_(j1,j2) ordered pairs (b₁, b₂) in D satisfying S₁ ⊆ b₁ and S₂ ⊆ b₂. In W. J. Martin [Designs in product association schemes, submitted for publication], Delsarte's theory of designs in association schemes is extended to products of Q-polynomial association schemes. Mixed t-designs arise as a particularly interesting case. These include symmetric designs with a distinguished block and α-resolvable balanced incomplete block designs as examples. The theory in the above-mentioned paper yields results on mixed t-designs analogous to those known for ordinary t-designs, such as the Ray-Chaudhuri/Wilson bound. For example, the analogue of Fisher's inequality gives |D| ≥ v₁ + v₂ − 1 for mixed 2-designs with Bose's condition on resolvable designs as a special case. Partial results are obtained toward a classification of those mixed 2-designs D with |D| = v₁ + v₂ − 1. The central result of this article is Theorem 3.1, an analogue of the Assmus–Mattson theorem which allows us to construct mixed (t + 1 − s)-designs from any t-design with s distinct block intersection numbers. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:151–163, 1998     

On qualitatively independent partitions and related problems

The paper is concerned with several related combinatorial problems one of which is that of estimating the numbers of qualitatively independent p-partitions. Besides nonconstructive basic estimates, a constructive procedure yielding not much worse ones is presented. In conclusion, some applications are shown.     

Modular Hadamard martrices and related designs

A square matrix with entries ± 1 is called a modular Hadamard matrix if the inner product of each two distinct row vectors is a multiple of some fixed (positive) integer. This paper initiates the study of modular Hadamard matrices and the combinatorial designs associated with them. The related combinatorial designs are the main concern of this paper; some results dealing with the existence and construction of modular Hadamard matrices will be included in a later paper.     

Constructions for resolvable and related designs

Methods are given for constructing block designs, using resolvable designs. These constructions yield methods for generating resolvable and affine designs and also affine designs with affine duals. The latter are transversal designs or semi-regular group divisible designs with ₁=0 whose duals are also designs of the same type and parameters. The paper is a survey of some old and some recent constructions.     

Generalized balanced tournament designs and related codes

A generalized balanced tournament design, or a GBTD(k, m) in short, is a (km, k, k − 1)-BIBD defined on a km-set V. Its blocks can be arranged into an m × (km − 1) array in such a way that (1) every element of V is contained in exactly one cell of each column, and (2) every element of V is contained in at most k cells of each row. In this paper, we present a new construction for GBTDs and show that a GBTD(4, m) exists for any integer m ≥ 5 with at most eight possible exceptions. A link between a GBTD(k, m) and a near constant composition code is also mentioned. The derived code is optimal in the sense of its size.      

Mixed partitions of PG(3,q)

A mixed partition of PG(2n−1,q²) is a partition of the points of PG(2n−1,q²) into (n−1)-spaces and Baer subspaces of dimension 2n−1. In (Bruck and Bose, J. Algebra 1 (1964) 85) it is shown that such a mixed partition of PG(2n−1,q²) can be used to construct a (2n−1)-spread of PG(4n−1,q) and hence a translation plane of order q²ⁿ. In this paper, we provide several new examples of such mixed partitions in the case when n=2.     

Modular Hadamard matrices and related designs,III

This paper continues the investigations presented in two previous papers on the same subject by the author and A. T. Butson. Modular Hadamard matrices havingn odd andh ≡ ? 1 (modn) are studied for a few values of the parametersn andh. Also, some results are obtained for the two related combinatorial designs. These results include: a discussion on the known techniques for constructing pseudo (v, k, λ)-designs; the fact that the existence of one of the two related designs always implies the existence of the other; and some information about the column sums of the incidence matrix of each of the two ‘maximal’ cases of (m, v, k ₁,λ ₁,k ₂,λ ₂,f, λ ₃)-designs.     

Doubly resolvable Steiner quadruple systems and related designs

Two resolutions of the same \(\hbox {SQS}(v)\) are said to be orthogonal, when each parallel class of one resolution has at most one block in common with each parallel class of the other resolution. If an \(\hbox {SQS}(v)\) has two orthogonal resolutions, the \(\hbox {SQS}(v)\) is called a doubly resolvable \(\hbox {SQS}(v)\). In this paper, we use a quadrupling construction to obtain an infinite class of doubly resolvable Steiner quadruple systems. We also give some results of doubly resolvable H designs and doubly resolvable candelabra quadruple systems.     

A generalization of combinatorial designs and related codes

In this paper we study a certain generalization of combinatorial designs related to almost difference sets, namely the t-adesign, which was coined by Ding (Codes from difference sets, 2015). It is clear that 2-adesigns are partially balanced incomplete block designs which naturally arise in many combinatorial and statistical problems. We discuss some of their basic properties and give several constructions of 2-adesigns (some of which correspond to new almost difference sets and some to new almost difference families), as well as two constructions of 3-adesigns. We discuss basic properties of the incidence matrices and make an initial investigation into the codes which they generate. We find that many of the codes have good parameters in the sense they are optimal or have relatively high minimum distance.     

Sequentially congruent partitions and partitions into squares

The Ramanujan Journal - In recent work, M. Schneider and the first author studied a curious class of integer partitions called “sequentiallyc congruent” partitions: the mth part is...     

Quasi-symmetric designs related to the triangular graph

LetT _m be the adjacency matrix of the triangular graph. We will give conditions for a linear combination ofT _m, I andJ to be decomposable. This leads to Bruck-Ryser-Chowla like conditions for, what we call, triangular designs. These are quasi-symmetric designs whose block graph is the complement of the triangular graph. For these designs our conditions turn out to be stronger than the known non-existence results for quasi-symmetric designs.