In a secret-sharing scheme, a secret is shared among n parties such that the secret can be recovered by authorized coalitions, while it should be kept hidden from unauthorized coalitions. In this work we study secret-sharing for k-slice access structures, in which coalitions of size k are either authorized or not, larger coalitions are authorized and smaller are unauthorized. Known schemes for these access structures had smaller shares for small k’s than for large ones; hence our focus is on “high” \((n-k)\) -slices where k is small. Our work is inspired by several motivations: 1) Obtaining efficient schemes (with perfect or computational security) for natural families of access structures; 2) Making progress in the search for better schemes for general access structures, which are often based on schemes for slice access structures; 3) Proving or disproving the conjecture by Csirmaz (J. Math. Cryptol., 2020) that an access structures and its dual can be realized by secret-sharing schemes with the same share size. The main results of this work are: