{"title":"汉明空间的另一个直径定理:最优群反码","authors":"R. Ahlswede","doi":"10.1109/ITW.2006.1633814","DOIUrl":null,"url":null,"abstract":"In the last century together with Levon Khachatrian we established a diametric theorem in Hamming space H<sup>n</sup>=(X<sup>n</sup>,d<inf>H</inf>). Now we contribute a diametric theorem for such spaces, if they are endowed with the group structure G<sup>n</sup>=<sup>n</sup>Σ<inf>1</inf>G, the direct sum of a group G on X={0,1,...,q-1}, and as candidates are considered subgroups of G<sup>n</sup>. For all finite groups G, every permitted distance d, and all n≥d subgroups of G<sup>n</sup>with diameter d have maximal cardinality q<sup>d</sup>. Other extremal problems can also be studied in this setting.","PeriodicalId":293144,"journal":{"name":"2006 IEEE Information Theory Workshop - ITW '06 Punta del Este","volume":"42 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2006-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Another diametric theorem in Hamming spaces: optimal group anticodes\",\"authors\":\"R. Ahlswede\",\"doi\":\"10.1109/ITW.2006.1633814\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In the last century together with Levon Khachatrian we established a diametric theorem in Hamming space H<sup>n</sup>=(X<sup>n</sup>,d<inf>H</inf>). Now we contribute a diametric theorem for such spaces, if they are endowed with the group structure G<sup>n</sup>=<sup>n</sup>Σ<inf>1</inf>G, the direct sum of a group G on X={0,1,...,q-1}, and as candidates are considered subgroups of G<sup>n</sup>. For all finite groups G, every permitted distance d, and all n≥d subgroups of G<sup>n</sup>with diameter d have maximal cardinality q<sup>d</sup>. Other extremal problems can also be studied in this setting.\",\"PeriodicalId\":293144,\"journal\":{\"name\":\"2006 IEEE Information Theory Workshop - ITW '06 Punta del Este\",\"volume\":\"42 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2006-03-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2006 IEEE Information Theory Workshop - ITW '06 Punta del Este\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/ITW.2006.1633814\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2006 IEEE Information Theory Workshop - ITW '06 Punta del Este","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/ITW.2006.1633814","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Another diametric theorem in Hamming spaces: optimal group anticodes
In the last century together with Levon Khachatrian we established a diametric theorem in Hamming space Hn=(Xn,dH). Now we contribute a diametric theorem for such spaces, if they are endowed with the group structure Gn=nΣ1G, the direct sum of a group G on X={0,1,...,q-1}, and as candidates are considered subgroups of Gn. For all finite groups G, every permitted distance d, and all n≥d subgroups of Gnwith diameter d have maximal cardinality qd. Other extremal problems can also be studied in this setting.
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