Mutually orthogonal binary frequency squares of mixed type
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917658655457280 |
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| author | Bodkin, Carly Wanless, Ian M. |
| author_facet | Bodkin, Carly Wanless, Ian M. |
| contents | A \emph{frequency square} is a matrix in which each row and column is a permutation of the same multiset of symbols. Two frequency squares $F_1$ and $F_2$ with symbol multisets $M_1$ and $M_2$ are \emph{orthogonal} if the multiset of pairs obtained by superimposing $F_1$ and $F_2$ is $M_1\times M_2$. A set of MOFS is a set of frequency squares in which each pair is orthogonal. We first generalise the classical bound on the cardinality of a set of MOFS to cover the case of \emph{mixed type}, meaning that the symbol multisets are allowed to vary between the squares in the set.
A frequency square is \emph{binary} if it only uses the symbols 0 and 1. We say that a set $\mathcal{F}$ of MOFS is \emph{type-maximal} if it cannot be extended to a larger set of MOFS by adding a square whose symbol multiset matches that of at least one square already in $\mathcal{F}$. Building on pioneering work by Stinson, several recent papers have found conditions that are sufficient to show that a set of binary MOFS is type-maximal. We generalise these papers in several directions, finding new conditions that imply type-maximality. Our results cover sets of binary frequency squares of mixed type. Also, where previous papers used parity arguments, we show the merit of arguments that use moduli greater than 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_01746 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mutually orthogonal binary frequency squares of mixed type Bodkin, Carly Wanless, Ian M. Combinatorics 05B15 A \emph{frequency square} is a matrix in which each row and column is a permutation of the same multiset of symbols. Two frequency squares $F_1$ and $F_2$ with symbol multisets $M_1$ and $M_2$ are \emph{orthogonal} if the multiset of pairs obtained by superimposing $F_1$ and $F_2$ is $M_1\times M_2$. A set of MOFS is a set of frequency squares in which each pair is orthogonal. We first generalise the classical bound on the cardinality of a set of MOFS to cover the case of \emph{mixed type}, meaning that the symbol multisets are allowed to vary between the squares in the set. A frequency square is \emph{binary} if it only uses the symbols 0 and 1. We say that a set $\mathcal{F}$ of MOFS is \emph{type-maximal} if it cannot be extended to a larger set of MOFS by adding a square whose symbol multiset matches that of at least one square already in $\mathcal{F}$. Building on pioneering work by Stinson, several recent papers have found conditions that are sufficient to show that a set of binary MOFS is type-maximal. We generalise these papers in several directions, finding new conditions that imply type-maximality. Our results cover sets of binary frequency squares of mixed type. Also, where previous papers used parity arguments, we show the merit of arguments that use moduli greater than 2. |
| title | Mutually orthogonal binary frequency squares of mixed type |
| topic | Combinatorics 05B15 |
| url | https://arxiv.org/abs/2212.01746 |