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Some randomized code constructions from group actions

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dc.contributor.author Bazzi L.M.J.
dc.contributor.author Mitter S.K.
dc.contributor.editor
dc.date 2006
dc.date.accessioned 2017-10-04T11:07:33Z
dc.date.available 2017-10-04T11:07:33Z
dc.date.issued 2006
dc.identifier 10.1109/TIT.2006.876244
dc.identifier.isbn
dc.identifier.issn 00189448
dc.identifier.uri http://hdl.handle.net/10938/14699
dc.description.abstract We study in this paper randomized constructions of binary linear codes that are invariant under the action of some group on the bits of the codewords. We study a non-Abelian randomized construction corresponding to the action of the dihedral group on a single copy of itself as well as a randomized Abelian construction based on the action of an Abelian group on a number of disjoint copies of itself. Cyclic codes have been extensively studied over the last 40 years. However, it is still an open question as to whether there exist asymptotically good binary cyclic codes. We argue that by using a slightly more complex group than a cyclic group, namely, the dihedral group, the existence of asymptotically good codes that are invariant under the action of the group on itself can be guaranteed. In particular, we show that, for infinitely many block lengths, a random ideal in the binary group algebra of the dihedral group is an asymptotically good rate-half code with a high probability. We argue also that a random code that is invariant under the action of an Abelian group G of odd order on k disjoint copies of itself satisfies the binary Gilbert-Varshamov (GV) bound with a high probability for rate 1-k under a condition on the family of groups. The underlying condition is in terms of the growth of the smallest dimension of a nontrivial F2-representation of the group and is satisfied by roughly most Abelian groups of odd order, and specifically by almost all cyclic groups of prime order. © 2006 IEEE.
dc.format.extent
dc.format.extent Pages: (3210-3219)
dc.language English
dc.publisher PISCATAWAY
dc.relation.ispartof Publication Name: IEEE Transactions on Information Theory; Publication Year: 2006; Volume: 52; no. 7; Pages: (3210-3219);
dc.relation.ispartofseries
dc.relation.uri
dc.source Scopus
dc.subject.other
dc.title Some randomized code constructions from group actions
dc.type Article
dc.contributor.affiliation Bazzi, L.M.J., Department of Electrical and Computer Engineering, American University of Beirut (AUB), Beirut 1107 2020, Lebanon
dc.contributor.affiliation Mitter, S.K., Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, United States
dc.contributor.authorAddress Bazzi, L.M.J.; Department of Electrical and Computer Engineering, American University of Beirut (AUB), Beirut 1107 2020, Lebanon; email: Louay.Bazzi@aub.edu.lb
dc.contributor.authorCorporate University: American University of Beirut; Faculty: Faculty of Engineering and Architecture; Department: Electrical and Computer Engineering;
dc.contributor.authorDepartment Electrical and Computer Engineering
dc.contributor.authorDivision
dc.contributor.authorEmail Louay.Bazzi@aub.edu.lb; mitter@mit.edu
dc.contributor.authorFaculty Faculty of Engineering and Architecture
dc.contributor.authorInitials Bazzi, LAJ
dc.contributor.authorInitials Mitter, SK
dc.contributor.authorOrcidID
dc.contributor.authorReprintAddress Bazzi, LAJ (reprint author), Amer Univ Beirut, Dept Elect and Comp Engn, Beirut 11072020, Lebanon.
dc.contributor.authorResearcherID
dc.contributor.authorUniversity American University of Beirut
dc.description.cited Burrow M., 1965, REPRESENTATION THEOR; CHEN CL, 1969, INFORM CONTROL, V15, P407, DOI 10.1016-S0019-9958(69)90497-5; CHEPYZHOV V, 1993, PROBL PEREDACHI INF, V28, P33; Curtis C.W., 1962, REPRESENTATION THEOR; KASAMI T, 1974, IEEE T INFORM THEORY, V20, P679, DOI 10.1109-TIT.1974.1055262; LIDL R, 1983, FINITE FIELDS NUMBER; LINT X, 1999, GRADUATE TEXTS MATH; LUBOTZKY A, 1988, COMBINATORICA, V8, P261, DOI 10.1007-BF02126799; MACWILLIAMS FJ, 1969, COMBINATORIAL MATH I, P317; MACWILLIAMS JF, 1992, THEORY ERROR CORRECT; Margulis G. A., 1988, Problems of Information Transmission, V24; MCDONALD, 1974, FINITE RINGS IDENTIT; PIRET P, 1985, IEEE T INFORM THEORY, V31, P520, DOI 10.1109-TIT.1985.1057061; Pless V.S., 1998, HDB CODING THEORY; Shparlinsky I. E., 1986, PROBLEMY PEREDACHI I, V22, P43; Sipser M, 1996, IEEE T INFORM THEORY, V42, P1710, DOI 10.1109-18.556667; WARD HN, 1974, J ALGEBRA, V29, P150, DOI 10.1016-0021-8693(74)90120-3
dc.description.citedCount 6
dc.description.citedTotWOSCount 5
dc.description.citedWOSCount 5
dc.format.extentCount 10
dc.identifier.articleNo
dc.identifier.coden IETTA
dc.identifier.pubmedID
dc.identifier.scopusID 33746891219
dc.identifier.url
dc.publisher.address 445 HOES LANE, PISCATAWAY, NJ 08855 USA
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dc.relation.ispartofConferenceTitle
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dc.relation.ispartOfISOAbbr IEEE Trans. Inf. Theory
dc.relation.ispartOfIssue 7
dc.relation.ispartOfPart
dc.relation.ispartofPubTitle IEEE Transactions on Information Theory
dc.relation.ispartofPubTitleAbbr IEEE Trans. Inf. Theory
dc.relation.ispartOfSpecialIssue
dc.relation.ispartOfSuppl
dc.relation.ispartOfVolume 52
dc.source.ID WOS:000238921200022
dc.type.publication Journal
dc.subject.otherAuthKeyword Abelian codes
dc.subject.otherAuthKeyword Dihedral group
dc.subject.otherAuthKeyword Group actions
dc.subject.otherAuthKeyword Group algebra
dc.subject.otherAuthKeyword Probabilistic method
dc.subject.otherAuthKeyword Quasi-cyclic codes
dc.subject.otherChemCAS
dc.subject.otherIndex Algebra
dc.subject.otherIndex Information theory
dc.subject.otherIndex Random processes
dc.subject.otherIndex Abelian codes
dc.subject.otherIndex Dihedral group
dc.subject.otherIndex Quasi-cyclic codes
dc.subject.otherIndex Randomized code constructions
dc.subject.otherIndex Binary codes
dc.subject.otherKeywordPlus QUASI-CYCLIC CODES
dc.subject.otherWOS Computer Science, Information Systems
dc.subject.otherWOS Engineering, Electrical and Electronic


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