Pairs of MOLS of order ten satisfying non-trivial relations

Michael J. Gill, Ian M. Wanless

Research output: Contribution to journalArticleResearchpeer-review


A relation on a k-net (n) (or, equivalently, a set of k- 2 mutually orthogonal Latin squares of order n) is an F2 linear dependence within the incidence matrix of the net. Dukes and Howard (2014) showed that any 6 -net (10 ) satisfies at least two non-trivial relations, and classified the relations that could appear in such a net. We find that, up to equivalence, there are 18526320 pairs of MOLS satisfying at least one non-trivial relation. None of these pairs extend to a triple. We also rule out one other relation on a set of 3-MOLS from Dukes and Howard’s classification.

Original languageEnglish
Pages (from-to)1293-1313
Number of pages21
JournalDesigns Codes and Cryptography
Issue number4
Publication statusPublished - Apr 2023


  • Frequency square
  • Latin square
  • MOLS
  • Net
  • Relation
  • Transversal

Cite this