Optimal codes in the Enomoto-Katona space

Coding in a new metric space, the Enomoto-Katona space, is considered recently in connection to the study of implication structures of functional dependencies and their generalizations in relational databases. The central problem here is the determination of C(n, k, d), the size of an optimal code of length n, weight k, and distance d in the Enomoto-Katona space. The value of C(n, k, d) is known only for some congruence classes of n when (k, d) ∈ {(2, 3), (3, 5)}. In this paper, we obtain new infinite families of optimal codes in the Enomoto-Katona space. In particular, C(n, k, 2k-1) is determined for all sufficiently large n satisfying either n ≡ 1 mod k and n(n-1) ≡ 0 mod 2k², or n ≡ 0 mod k.