Combinatorics of Singly-Repairable Families

A non-empty set F of n-bit vectors over alphabet {0, 1} is called singly repairable, if every vector u ∈ F satisfies the following conditions: (i) if any bit of u is changed (from 0 to 1 or vice versa), the new vector does not belong to F (ii) there is a unique choice of a different bit that can then be changed to give another vector 6= u in F . Such families F exist only for even n and we show that 2n/2 ≤ |F| ≤ 2n+1 (n+2) . The lower bound is tight for all even n and we show that the families of this size are unique under a natural notion of isomorphism (namely, translations and permutation of coordinates). We also construct families that achieve the upper bound when n is of the form 2m − 2. For general even n, we construct families of size at least 2n/n. Of particular interest are minimal singly-repairable families. We show that such families have size at most 2n/n and we construct families achieving this upper bound when n is a power of 2. For general even n, we construct minimal families of size Ω(2n/n2). The study of these families was inspired by a computational scheduling problem.