{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-I3WC4A8L/11876778-df07-4662-9932-c828470a3291/PDF","dcterms:extent":"318 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-I3WC4A8L/b62cbe5d-8a53-4b18-a685-7e000aedb31e/TEXT","dcterms:extent":"17 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-I3WC4A8L","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2022","dc:creator":"Mollard, Michel","dc:format":[{"@xml:lang":"sl","#text":"letnik:22"},{"@xml:lang":"sl","#text":"številka:3"},{"@xml:lang":"sl","#text":"P3.10 (7 str.)"}],"dc:identifier":["DOI:10.26493/1855-3974.2308.de6","COBISSID_HOST:118549507","ISSN:1855-3966","URN:URN:NBN:SI:doc-I3WC4A8L"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"en","#text":"error correcting codes"},{"@xml:lang":"en","#text":"Fibonacci cube"},{"@xml:lang":"sl","#text":"Fibonaccijeva kocka"},{"@xml:lang":"sl","#text":"kode za popravljanje napak"},{"@xml:lang":"en","#text":"perfect code"},{"@xml:lang":"sl","#text":"popolna koda"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"The (non-)existence of perfect codes in Lucas cubes|"},"dc:description":[{"@xml:lang":"sl","#text":"The Fibonacci cube of dimension ?$n$?, denoted as ?$\\Gamma_n$?, is the subgraph of the ?$n$?-cube ?$Q_n$? induced by vertices with no consecutive 1's. Ashrafi and his co-authors proved the non-existence of perfect codes in ?$\\Gamma_n$? for ?$n \\geq 4$?. As an open problem the authors suggest to consider the existence of perfect codes in generalizations of Fibonacci cubes. The most direct generalization is the family ?$\\Gamma_n (1^s)$? of subgraphs induced by strings without ?$1^s$? as a substring where ?$s \\geq 2$? is a given integer. In a precedent work we proved the existence of a perfect code in ?$\\Gamma_n (1^s)$? for ?$n = 2^p - 1$? and ?$s \\geq 3.2^{p-2}$? for any integer ?$p \\geq 2$?. The Lucas cube ?$\\Lambda_n$? is obtained from ?$\\Gamma_n$? by removing vertices that start and end with 1. Very often the same problems are studied on Fibonacci cubes and Lucas cube. In this note we prove the non-existence of perfect codes in ?$\\Lambda_n$? for ?$n \\geq 4$? and prove the existence of perfect codes in some generalized Lucas cube ?$\\Lambda_n (1^s)$?"},{"@xml:lang":"sl","#text":"Fibonaccijeva kocka dimenzije ?$n$?, označena z ?$\\Gamma_n$?, je podgraf ?$n$?-kocke ?$Q_n$?, ki je induciran z vozlišči brez zaporednih 1. Ashrafi in njegovi soavtorji so dokazali neobstoj popolnih kod v ?$\\Gamma_n$? za ?$n \\geq 4$?. Kot odprt problem predlagajo obravnavo obstoja popolnih kod v posplošitvah Fibonaccijevih kock. Najbolj neposredna posplošitev je družina ?$\\Gamma_n (1^s)$? podgrafov, induciranih z nizi, ki so brez ?$1^s$? kot podniza, pri čemer je ?$s \\geq 2$? dano celo število. V prejšnjem članku smo dokazali obstoj popolne kode v ?$\\Gamma_n (1^s)$? za ?$n = 2^p - 1$? in ?$s \\geq 3.2^{p-2}$? za vsako celo število ?$p \\geq 2$?. Lucasovo kocko ?$\\Lambda_n$? dobimo iz ?$\\Gamma_n$? z odstranitvijo vozlišč, ki se začnejo in končajo z 1. Pogosto so isti problemi preučevani na Fibonaccijevih kockah in na Lucasovi kocki. V tem kratkem članku dokažemo neobstoj popolnih kod v ?$\\Lambda_n$? za ?$n \\geq 4$? in dokažemo eksistenco popolnih kod v nekaterih posplošenih Lucasovih kockah ?$\\Lambda_n (1^s)$?"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-I3WC4A8L","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-I3WC4A8L"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-I3WC4A8L/11876778-df07-4662-9932-c828470a3291/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-I3WC4A8L/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-I3WC4A8L"}}}}