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There are ten configurations of two bowties that can arise in a bowtie system. The avoidance spectrum for three of these was determined in a previous paper (Aequat. Math. 85 (2013), 347-358). In this paper the avoidance spectrum for a further five configurations is de-termined.

A lower bound is presented for the minimal number of filled cells in a maximal partial Latin hypercube of dimension d and order n. The result generalises and extends previous results for d = 2 (Latin squares) and d = 3 (Latin cubes). Explicit constructions show that this bound is near-optimal for large n > d. For d > n, a connection with Hamming codes shows that this lower bound gives a related upper bound for the same quantity. The results can be interpreted in terms of independent dominating sets in certain graphs, and in terms of codes that have covering radius 1 and minimum distance at least 2.

There are ten configurations of two bowties that can arise in a bowtie system. The avoidance spectrum for three of these was determined in a previous paper (Aequat. Math. 85 (2013), 347-358). In this paper the avoidance spectrum for a further five configurations is de-termined.

Denote by $\mathcal{A}_p^k$ the Latin square of order $n=p^k$ formed by the Cayley table of the additive group $(\mathbb{Z}_p^k,+)$, where $p$ is an odd prime and $k$ is a positive integer. It is shown that for each $p$ there exists $Q>0$ such that for all sufficiently large $k$, the number of transversals in $\mathcal{A}_p^k$ exceeds $(nQ)^{\frac{n}{p(p-1)}}$.