Secret sharing schemes are one of the most important primitives in distributed systems. In perfect secret sharing schemes, collaboration between unauthorized participants cannot reduce their uncertainty about the secret. This paper presents a perfect secret sharing scheme arising from critical sets of Room squares.

Let \(L(n, k, p, t)\) denote the minimum number of subsets of size \(k\) (\(k\)-subsets) of a set of size \(n\) (\(n\)-set) such that any \(p\)-subset intersects at least one of these \(k\)-subsets in at least \(t\) elements. The value of \(L(n, 6, 6, 2)\) is determined for \(n \leq 54\).

The structure of cocyclic Hadamard matrices allows for a much faster and more systematic search for binary, self-dual codes. Here, we consider \(\mathbf{Z}_{2}^{2} \times \mathbf{Z}_{t}\)-cocyclic Hadamard matrices for \(t = 3, 5, 7,\) and \(9\) to yield binary
self-dual codes of lengths \(24, 40, 56,\) and \(72\). We show that the extended Golay code cannot be obtained as a member of this class and also demonstrate the existence of four apparently new codes – a \([56, 28, 8]\) code and three \([72, 36, 8]\) codes.