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Maximal sets of mutually orthogonal frequency squares and Doehlert-Klee designs

Carly Bodkin, Nicholas J. Cavenagh, Ian M. Wanless

TL;DR

The paper addresses constructing large sets of binary MOFS and, in particular, type-maximal MOFS for various orders. It establishes a precise equivalence between binary MOFS and a special class of Doehlert-Klee designs, enabling MOFS to be generated or detected via DK-design structure; it also develops cyclic-generation techniques and large-scale constructions. Key contributions include new DK-based constructions for type-maximal MOFS, cyclic and dilation-based methods, and explicit large examples (including $V=12615$ with complex block decompositions) that push the known boundaries. These results deepen the connection between MOFS and DK-design theory, offering practical pathways to generate large maximal systems and highlighting open questions in odd-order MOFS and mixed-type sum behavior.

Abstract

A binary frequency square of type $(n;λ_0,λ_1)$ is a $(0,1)$-matrix of order $n$ with $λ_0$ zeros and $λ_1$ ones in each row and in each column. Two such squares are orthogonal if there are exactly $λ_1^2$ cells where both squares contain ones. A set of binary MOFS is a set of binary frequency squares in which each pair is orthogonal. A set of binary MOFS of type $(n;λ_0,λ_1)$ is type maximal if there is no square of the type $(n;λ_0,λ_1)$ that is orthogonal to every square in the set. A Doehlert-Klee design consists of points $V$ and blocks $B$, where every pair of points occurs in precisely $Λ$ blocks and every point occurs in precisely $R$ blocks, where $R^2=Λ|B|$. We show that sets of binary MOFS are equivalent to a particular kind of Doehlert-Klee design. In a distinct application, Doehlert-Klee designs can also be used to construct sets of binary MOFS that are cyclically generated from their first rows. We use these connections to find new constructions for sets of type-maximal binary MOFS.

Maximal sets of mutually orthogonal frequency squares and Doehlert-Klee designs

TL;DR

The paper addresses constructing large sets of binary MOFS and, in particular, type-maximal MOFS for various orders. It establishes a precise equivalence between binary MOFS and a special class of Doehlert-Klee designs, enabling MOFS to be generated or detected via DK-design structure; it also develops cyclic-generation techniques and large-scale constructions. Key contributions include new DK-based constructions for type-maximal MOFS, cyclic and dilation-based methods, and explicit large examples (including with complex block decompositions) that push the known boundaries. These results deepen the connection between MOFS and DK-design theory, offering practical pathways to generate large maximal systems and highlighting open questions in odd-order MOFS and mixed-type sum behavior.

Abstract

A binary frequency square of type is a -matrix of order with zeros and ones in each row and in each column. Two such squares are orthogonal if there are exactly cells where both squares contain ones. A set of binary MOFS is a set of binary frequency squares in which each pair is orthogonal. A set of binary MOFS of type is type maximal if there is no square of the type that is orthogonal to every square in the set. A Doehlert-Klee design consists of points and blocks , where every pair of points occurs in precisely blocks and every point occurs in precisely blocks, where . We show that sets of binary MOFS are equivalent to a particular kind of Doehlert-Klee design. In a distinct application, Doehlert-Klee designs can also be used to construct sets of binary MOFS that are cyclically generated from their first rows. We use these connections to find new constructions for sets of type-maximal binary MOFS.

Paper Structure

This paper contains 9 sections, 16 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Designs
  4. Mutually orthogonal frequency squares
  5. Binary MOFS are Doehlert-Klee designs
  6. Maximal sets of MOFS from Doehlert-Klee designs
  7. An approach using cyclic frequency squares
  8. Large examples
  9. Discussion

Key Result

Theorem 2.1

Let $w$ be a non-negative integer. Let ${\mathcal{F}}=\{F_1,F_2,\dots,F_k\}$ be a set of binary MOFS where $F_t$ has type $(n;\lambda_{0,t},\lambda_{1,t})$ for $1 \leqslant t \leqslant k$. Suppose ${\mathcal{F}}$ has $\mathbb{Z}_w$-sum that, up to permutation of the rows and columns, has the followi where $x_1,x_2,x_3,x_4$ are non-negative integers satisfying $x_1+x_4\equiv x_2+x_3 \bmod w$. Let $

Theorems & Definitions (25)

  • Theorem 2.1
  • Corollary 2.2
  • Example 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 15 more