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Optimal Codes in the Enomoto-Katona Space

Published online by Cambridge University Press:  09 October 2014

YEOW MENG CHEE ,
HAN MAO KIAH ,
HUI ZHANG  and
XIANDE ZHANG
YEOW MENG CHEE
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore (e-mail: ymchee@ntu.edu.sg, huizhang@ntu.edu.sg, XiandeZhang@ntu.edu.sg)
HAN MAO KIAH
Affiliation:
Coordinated Science Lab, University of Illinois, Urbana-Champaign, 1308 W. Main Street, Urbana, IL 61801, USA (e-mail: hmkiah@illinois.edu)
HUI ZHANG
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore (e-mail: ymchee@ntu.edu.sg, huizhang@ntu.edu.sg, XiandeZhang@ntu.edu.sg)
XIANDE ZHANG
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore (e-mail: ymchee@ntu.edu.sg, huizhang@ntu.edu.sg, XiandeZhang@ntu.edu.sg)
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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 2k2, 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.

Keywords

Primary 05C35Secondary 05C1557M25
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Issue 2 , March 2015 , pp. 382 - 406
Copyright
Copyright © Cambridge University Press 2014 

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