A lower bound on HMOLS with equal sized holes [article]

Michael Bailey, Coen del valle, Peter J. Dukes
2020 arXiv   pre-print
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
more » ... clotomic 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^n) ≥ (log n)^1/δ for any δ>2 and all n > n_0(h,δ).
arXiv:2008.08788v1 fatcat:nq7ot5kqlrfovmlw5xilaa2uzm