Optimal codes in the Enomoto-Katona space

Abstract

Coding in a new metric space, called the Enomoto-Katona space, has recently been considered in connection with the study of implication structures of functional dependencies and their generalizations in relational databases. The central problem 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) was 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 and verify a conjecture of Brightwell and Katona in certain instances. 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 2k 2 , or n ≡ 0 mod k. We also give complete solutions for k = 2 and determine C(n, 3, 5) for certain congruence classes of n with finite exceptions.

Applications of graph containers in the Boolean lattice

“…Recently the value of has been determined for many values of ( n, k, d ) (see ). As an example, we have equality or are ‘close’ to equality in () when and for certain (congruency) classes of n (see ). Further, the bound in () is asymptotically sharp for fixed k, d and (see ).…”

“…As an example, we have equality or are 'close' to equality in (5.3) when k ≥ 2, d = 2k−1 and for certain (congruency) classes of n (see [12]). Further, the bound in (5.3) is asymptotically sharp for fixed k, d and n → ∞ (see [9]).…”