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