Row-column factorial designs and mutually orthogonal frequency rectangles

A (full) qᵏ factorial design with replication λ is the multi-set containing all possible q-ary sequences of length k, each occurring exactly λ times. An m × n row-column factorial design is any arrangement of λ replicates of the qᵏ factorial design in an m × n array. We say that the design has strength t if each row and column is an orthogonal array of strength t. We denote such a design by Iₖ (m,n,q,t). A frequency rectangle of type FR(m,n;q) is an m × n array based on a symbol set S of size q, such that each element of S appears exactly n/q times in each row and m/q times in each column. Two frequency rectangles of the same type are said to be orthogonal if each possible pair of symbols appears the same number of times when the two arrays are superimposed. By k–MOFR(m,n;q) we mean a set of k frequency rectangles of type FR(m,n;q) in which every pair is orthogonal. In Chapter 4, we give the necessary and sufficient conditions when a row-column factorial design of strength 1 exists. We show that an array of type Iₖ (m,n,q,1) exists if and only if (a) q|m, q|n and qᵏ|mn; (b) (k,q,m,n) ≠ (2,6,6,6) and (c) if (k,q,m) = (2,2,2) then 4 divides n. In Chapter 5, we discuss designs of strength 2 and above. We solve the case completely when t = 2 and q is a prime power: we show that there exists an array of type Iₖ(qᴹ,qᴺ,q,2) if and only if k ≤ M + N, k ≤ (qᴹ - 1)/(q - 1) and (k,M,q) ≠ (3,2,2). We also show that Iₖ+α(2αb,2ᵏ,2,2) exists whenever α ≥ 2 and 2α + α + 1 ≤ k < 2αb - α, assuming there exists a Hadamard matrix of order 4b. For strength 3 we restrict ourselves to the binary case, solving it completely when q is a power of 2. In Chapter 6, our focus is on mutually orthogonal frequency rectangles (MOFR). We use orthogonal arrays and Hadamard matrices to construct sets of MOFR. We also describe a new form of orthogonality for a set of frequency rectangles. We say that a k–MOFR(m,n;q) is t–orthogonal if each subset of size t, when superimposed, forms a qᵗ factorial design with replication mn/qᵗ. A set of vectors over a finite field is said to be t-independent if each subset of size t is linearly independent. We describe a relationship between a set of t–orthogonal MOFR and a set of t-independent vectors. We use known results from coding theory and related literature to formulate a table for the size of a set of t-independent vectors of length N ≤ 16, over F₂. We also describe a method to construct a set of (p - 1)–MOFR(2p,2p;2) where p is an odd prime, improving known lower bounds for all p ≥ 19

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Decompositions into spanning rainbow structures

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares and has been the focus of extensive research ever since. Euler posed a problem equivalent to finding properly n-edge-coloured complete bipartite graphs Kn,n which can be decomposed into rainbow perfect matchings. While there are proper edge-colourings of Kn,n without even a single rainbow perfect matching, the theme of this paper is to show that with some very weak additional constraints one can find many disjoint rainbow perfect matchings. In particular, we prove that if some fraction of the colour classes have at most (1−o(1))n edges then one can nearly-decompose the edges of Kn,n into edge-disjoint perfect rainbow matchings. As an application of this, we establish in a very strong form a conjecture of Akbari and Alipour and asymptotically prove a conjecture of Barat and Nagy. Both these conjectures concern rainbow perfect matchings in edge-colourings of Kn,n with quadratically many colours. Using our techniques, we also prove a number of results on near-decompositions of graphs into other rainbow structures like Hamiltonian cycles and spanning trees. Most notably, we prove that any properly coloured complete graph can be nearly-decomposed into spanning rainbow trees. This asymptotically proves the Brualdi-Hollingsworth and Kaneko-Kano-Suzuki conjectures which predict that a perfect decomposition should exist under the same assumptions

On the maximality of a set of mutually orthogonal Sudoku Latin Squares

The maximum number of mutually orthogonal Sudoku Latin squares (MOSLS) of order is n = m(2) is n - m. In this paper, we construct for n = q(2), q a prime power, a set of q(2) - q - 1 MOSLS of order q(2) that cannot be extended to a set of q(2) - q MOSLS. This contrasts to the theory of ordinary Latin squares of order n, where each set of n - 2 mutually orthogonal Latin Squares (MOLS) can be extended to a set of n - 1 MOLS (which is best possible). For this proof, we construct a particular maximal partial spread of size q(2) - q + 1 in PG(3,q) and use a connection between Sudoku Latin squares and projective geometry, established by Bailey, Cameron and Connelly

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science