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 n ) ≥ (log n) 1/δ for any δ > 2 and all n > n 0 (h, δ).

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 n ) ≥ (log n) 1/δ for any δ > 2 and all n > n 0 (h, δ).

Given a square matrix A over the integers, we consider the Z-module M A generated by the set of all matrices that are permutation-similar to A. Motivated by analogous problems on signed graph decompositions and block designs, we are interested in the completely symmetric matrices aI + bJ belonging to M A . We give a relatively fast method to compute a generator for such matrices, avoiding the need for a very large canonical form over Z. We consider several special cases in detail. In particular, the problem for symmetric matrices answers a question of Cameron and Cioabǎ on determining the eventual period for integers λ such that the λ-fold complete graph λKn has an edge-decomposition into a given (multi)graph.