Equidistant families of sets Original Research Article

A family of subsets of the set {1, 2, …, v} is called equidistant if the symmetric difference of any two distinct sets from the family has constant cardinality. Such families can be interpreted as binary equidistant codes of length v. It is known that for v ≡ 3 (mod 4), the maximum size of an equidistant code of length v is v + 1 and existence of a code of length v + 1 is equivalent to existence of a Hadamard matrix of order v + 1. We prove for any v that binary equidistant codes of size v exist if and only if there exist certain combinatorial designs. We specify these designs in case of binary equidistant codes of length v and size v having maximal distance between codewords, thereby extending results of Stinson and van Rees and van Lint.