{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,13]],"date-time":"2023-09-13T05:14:45Z","timestamp":1694582085504},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,10,9]],"date-time":"2014-10-09T00:00:00Z","timestamp":1412812800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2015,3]]},"abstract":"<jats:p>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<jats:italic>C<\/jats:italic>(<jats:italic>n,k,d<\/jats:italic>), the size of an optimal code of length<jats:italic>n<\/jats:italic>, weight<jats:italic>k<\/jats:italic>, and distance<jats:italic>d<\/jats:italic>in the Enomoto-Katona space. The value of<jats:italic>C<\/jats:italic>(<jats:italic>n,k,d<\/jats:italic>) was known only for some congruence classes of<jats:italic>n<\/jats:italic>when (<jats:italic>k,d<\/jats:italic>) \u2208 {(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,<jats:italic>C<\/jats:italic>(<jats:italic>n<\/jats:italic>,<jats:italic>k<\/jats:italic>, 2<jats:italic>k<\/jats:italic>\u2212 1) is determined for all sufficiently large<jats:italic>n<\/jats:italic>satisfying either<jats:italic>n<\/jats:italic>\u2261 1 mod<jats:italic>k<\/jats:italic>and<jats:italic>n<\/jats:italic>(<jats:italic>n<\/jats:italic>\u2212 1) \u2261 0 mod 2<jats:italic>k<\/jats:italic><jats:sup>2<\/jats:sup>, or<jats:italic>n<\/jats:italic>\u2261 0 mod<jats:italic>k<\/jats:italic>. We also give complete solutions for<jats:italic>k<\/jats:italic>= 2 and determine<jats:italic>C<\/jats:italic>(<jats:italic>n<\/jats:italic>,3,5) for certain congruence classes of<jats:italic>n<\/jats:italic>with finite exceptions.<\/jats:p>","DOI":"10.1017\/s0963548314000509","type":"journal-article","created":{"date-parts":[[2014,10,9]],"date-time":"2014-10-09T07:38:57Z","timestamp":1412840337000},"page":"382-406","source":"Crossref","is-referenced-by-count":1,"title":["Optimal Codes in the Enomoto-Katona Space"],"prefix":"10.1017","volume":"24","author":[{"given":"YEOW MENG","family":"CHEE","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"HAN MAO","family":"KIAH","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"HUI","family":"ZHANG","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"XIANDE","family":"ZHANG","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,10,9]]},"reference":[{"key":"S0963548314000509_ref20","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-23441-5_6"},{"key":"S0963548314000509_ref18","first-page":"121","article-title":"Combinatorial problems in the Enomoto\u2013Katona space","volume":"46","author":"Quistorff","year":"2009","journal-title":"Studia Sci. Math. Hungar."},{"key":"S0963548314000509_ref16","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2004.03.004"},{"key":"S0963548314000509_ref15","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/019\/0321765"},{"key":"S0963548314000509_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/S0378-3758(98)00029-9"},{"key":"S0963548314000509_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(92)90027-8"},{"key":"S0963548314000509_ref11","first-page":"213","article-title":"Minimal representations of branching dependencies","volume":"60","author":"Demetrovics","year":"1995","journal-title":"Acta Sci. Math."},{"key":"S0963548314000509_ref6","unstructured":"Bollob\u00e1s B. , F\u00fcredi Z. , Kantor I. , Katona G. O. H. and Leader, I. A coding problem for pairs of subsets. Preprint."},{"key":"S0963548314000509_ref5","doi-asserted-by":"publisher","DOI":"10.1145\/320493.320489"},{"key":"S0963548314000509_ref2","first-page":"160","volume-title":"The CRC Handbook of Combinatorial Designs","author":"Abel","year":"2007"},{"key":"S0963548314000509_ref1","first-page":"247","volume-title":"The CRC Handbook of Combinatorial Designs","author":"Abel","year":"2007"},{"key":"S0963548314000509_ref4","unstructured":"Beeri C. , Fagin R. and Howard J. H. (1977) A complete axiomatization for fuzzy functional and multivalued dependencies in fuzzy database relations. In Proc. ACM SIGMOD International Conference on Management of Data, pp. 47\u201361."},{"key":"S0963548314000509_ref8","unstructured":"Chee Y. M. , Kiah H. M. , Zhang H. and Zhang X. Addendum to optimal codes in the Enomoto\u2013Katona space. http:\/\/sites.google.com\/kiahhanmao\/optimalek"},{"key":"S0963548314000509_ref7","first-page":"139","article-title":"A new type of coding problem","volume":"38","author":"Brightwell","year":"2001","journal-title":"Studia Sci. Math. Hungar."},{"key":"S0963548314000509_ref17","doi-asserted-by":"publisher","DOI":"10.1006\/jcta.1999.3005"},{"key":"S0963548314000509_ref3","unstructured":"Armstrong W. W. (1974) Dependency structures of data base relationships. In Proc. IFIP Congress 1974, pp. 580\u2013583."},{"key":"S0963548314000509_ref19","doi-asserted-by":"publisher","DOI":"10.1145\/320576.320580"},{"key":"S0963548314000509_ref9","doi-asserted-by":"publisher","DOI":"10.1109\/ISIT.2013.6620240"},{"key":"S0963548314000509_ref14","doi-asserted-by":"crossref","DOI":"10.37236\/1606","article-title":"Pairs of disjoint q-element subsets far from each other","volume":"8","author":"Enomoto","year":"2001","journal-title":"Electron. J. Combin."},{"key":"S0963548314000509_ref21","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(72)90028-3"},{"key":"S0963548314000509_ref10","first-page":"377","article-title":"A relational model of data for large shared data banks","volume":"13","author":"Codd","year":"1970","journal-title":"Commun. Assoc. Comput. 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