## Abstract

It is known that *N*(*n*), the maximum number of mutually orthogonal latin squares of order n, satisfies the lower bound *N*(*n*)≥*n*^{1/14.8} for large *n*. For *h*≥2, relatively little is known about the quantity *N*(*h ^{n}*), which denotes the maximum number of ‘HMOLS’ or mutually orthogonal latin squares having a common equipartition into

*n*holes of a fixed size

*h*. We generalize a difference matrix method that had been used previously for explicit constructions of HMOLS. An estimate of R.M. Wilson on higher cyclotomic numbers guarantees our construction succeeds in suitably large finite fields. Feeding this into a generalized product construction, we are able to establish the lower bound

*N*(

*h*)≥(log

^{n}*n*)

^{1/δ}for any

*δ*>2 and all

*n*>

*n*(

_{0}*h*,

*δ*).

Original language | English |
---|---|

Article number | 101866 |

Number of pages | 16 |

Journal | Finite Fields and Their Applications |

Volume | 74 |

Early online date | 6 May 2021 |

DOIs | |

Publication status | Published - 1 Sept 2021 |

## Keywords

- Cyclotomy
- HMOLS
- Orthogonal latin squares
- Transversal design