A lower bound on HMOLS with equal sized holes

Michael Bailey, Coen del Valle, Peter J. Dukes*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


It is known that N(n), the maximum number of mutually orthogonal latin squares of order n, satisfies the lower bound N(n)≥n1/14.8 for large n. For h≥2, relatively little is known about the quantity N(hn), 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(hn)≥(log ⁡n)1/δ for any δ>2 and all n>n0(h,δ).

Original languageEnglish
Article number101866
Number of pages16
JournalFinite Fields and Their Applications
Early online date6 May 2021
Publication statusPublished - 1 Sept 2021


  • Cyclotomy
  • Orthogonal latin squares
  • Transversal design


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