Asymptotic Improvement of the Gilbert-Varshamov Bound for Linear Codes
The Gilbert-Varshamov (GV) bound states that the maximum size A 2 (n, d) of a binary code of length n and minimum distance d satisfies A 2 (n, d)ges2 n /V(n, d-1) where V(n, d)=Sigma i=0 d ( i n ) stands for the volume of a Hamming ball of radius d. Recently, Jiang and Vardy showed that for binary n...
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| Veröffentlicht in: | IEEE transactions on information theory 2008-09, Vol.54 (9), p.3865-3872 |
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| creator | Gaborit, P. Zemor, G. |
| description | The Gilbert-Varshamov (GV) bound states that the maximum size A 2 (n, d) of a binary code of length n and minimum distance d satisfies A 2 (n, d)ges2 n /V(n, d-1) where V(n, d)=Sigma i=0 d ( i n ) stands for the volume of a Hamming ball of radius d. Recently, Jiang and Vardy showed that for binary nonlinear codes this bound can be improved to A 2 (n, d)gescn2 n /(V(n, d-1)) for c a constant and d/nges0.499. In this paper, we show that certain asymptotic families of linear binary [n, n/2] random double circulant codes satisfy the same improved GV bound. |
| doi_str_mv | 10.1109/TIT.2008.928288 |
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Recently, Jiang and Vardy showed that for binary nonlinear codes this bound can be improved to A 2 (n, d)gescn2 n /(V(n, d-1)) for c a constant and d/nges0.499. In this paper, we show that certain asymptotic families of linear binary [n, n/2] random double circulant codes satisfy the same improved GV bound.</description><identifier>ISSN: 0018-9448</identifier><identifier>EISSN: 1557-9654</identifier><identifier>DOI: 10.1109/TIT.2008.928288</identifier><identifier>CODEN: IETTAW</identifier><language>eng</language><publisher>New York, NY: IEEE</publisher><subject>Applied sciences ; Asymptotic properties ; Binary codes ; Binary system ; Codes ; Coding, codes ; Computer Science ; Double circulant code ; Exact sciences and technology ; Gilbert-Varshamov (GV) bound ; Graph theory ; H infinity control ; Hamming distance ; Information processing ; Information Theory ; Information, signal and communications theory ; Linear code ; Mathematics ; Nonlinear systems ; Nonlinearity ; random coding ; Signal and communications theory ; Stands ; Supports ; Telecommunications and information theory ; Welding</subject><ispartof>IEEE transactions on information theory, 2008-09, Vol.54 (9), p.3865-3872</ispartof><rights>2008 INIST-CNRS</rights><rights>Copyright Institute of Electrical and Electronics Engineers, Inc. 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Recently, Jiang and Vardy showed that for binary nonlinear codes this bound can be improved to A 2 (n, d)gescn2 n /(V(n, d-1)) for c a constant and d/nges0.499. In this paper, we show that certain asymptotic families of linear binary [n, n/2] random double circulant codes satisfy the same improved GV bound.</description><subject>Applied sciences</subject><subject>Asymptotic properties</subject><subject>Binary codes</subject><subject>Binary system</subject><subject>Codes</subject><subject>Coding, codes</subject><subject>Computer Science</subject><subject>Double circulant code</subject><subject>Exact sciences and technology</subject><subject>Gilbert-Varshamov (GV) bound</subject><subject>Graph theory</subject><subject>H infinity control</subject><subject>Hamming distance</subject><subject>Information processing</subject><subject>Information Theory</subject><subject>Information, signal and communications theory</subject><subject>Linear code</subject><subject>Mathematics</subject><subject>Nonlinear systems</subject><subject>Nonlinearity</subject><subject>random coding</subject><subject>Signal and communications theory</subject><subject>Stands</subject><subject>Supports</subject><subject>Telecommunications and information theory</subject><subject>Welding</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2008</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNp9kT1v2zAQhomiAeqmmTt0EQqkRQc5PIqkyNE1msSAgSxOVoKljrACSXRI2UD-fSgo8NCh04G85z7eewn5CnQJQPXNbrNbMkrVUjPFlPpAFiBEXWop-EeyoBRUqTlXn8jnlJ7zkwtgC3K7Sq_9YQxj64pNf4jhhD0OYxF8Me6xuGu7vxjH8snGtLd9OBW_w3FoCh9isW0HtLFYhwbTF3LhbZfw6j1eksfbP7v1fbl9uNusV9vSCcbGUlc5Ouo1Stko4RsQStSV065BbpF562uuOFjOuQRASmkjvayYREedlNUl-TX33dvOHGLb2_hqgm3N_Wprpr9JJ_AaTpDZnzObVb0cMY2mb5PDrrMDhmMyGrSueCVYJn_8l6x4TSvJpvHf_wGfwzEOWbEBLTQTwKe5NzPkYkgpoj8vCtRMVplslZmsMrNVueL6va1NznY-2sG16VzGqIR8pWnPbzPXIuI5zSVVOt_mDYm3mYs</recordid><startdate>20080901</startdate><enddate>20080901</enddate><creator>Gaborit, P.</creator><creator>Zemor, G.</creator><general>IEEE</general><general>Institute of Electrical and Electronics Engineers</general><general>The Institute of Electrical and Electronics Engineers, Inc. (IEEE)</general><scope>97E</scope><scope>RIA</scope><scope>RIE</scope><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7SP</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>F28</scope><scope>FR3</scope><scope>1XC</scope><orcidid>https://orcid.org/0000-0002-6041-9554</orcidid></search><sort><creationdate>20080901</creationdate><title>Asymptotic Improvement of the Gilbert-Varshamov Bound for Linear Codes</title><author>Gaborit, P. ; Zemor, G.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c522t-93c52c0f9e66d85fd158573c9cde4ae2faf74841a444611e000d6f6326ec0c663</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2008</creationdate><topic>Applied sciences</topic><topic>Asymptotic properties</topic><topic>Binary codes</topic><topic>Binary system</topic><topic>Codes</topic><topic>Coding, codes</topic><topic>Computer Science</topic><topic>Double circulant code</topic><topic>Exact sciences and technology</topic><topic>Gilbert-Varshamov (GV) bound</topic><topic>Graph theory</topic><topic>H infinity control</topic><topic>Hamming distance</topic><topic>Information processing</topic><topic>Information Theory</topic><topic>Information, signal and communications theory</topic><topic>Linear code</topic><topic>Mathematics</topic><topic>Nonlinear systems</topic><topic>Nonlinearity</topic><topic>random coding</topic><topic>Signal and communications theory</topic><topic>Stands</topic><topic>Supports</topic><topic>Telecommunications and information theory</topic><topic>Welding</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Gaborit, P.</creatorcontrib><creatorcontrib>Zemor, G.</creatorcontrib><collection>IEEE All-Society Periodicals Package (ASPP) 2005-present</collection><collection>IEEE All-Society Periodicals Package (ASPP) 1998-Present</collection><collection>IEEE Xplore</collection><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Electronics & Communications Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>ANTE: Abstracts in New Technology & Engineering</collection><collection>Engineering Research Database</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>IEEE transactions on information theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Gaborit, P.</au><au>Zemor, G.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Asymptotic Improvement of the Gilbert-Varshamov Bound for Linear Codes</atitle><jtitle>IEEE transactions on information theory</jtitle><stitle>TIT</stitle><date>2008-09-01</date><risdate>2008</risdate><volume>54</volume><issue>9</issue><spage>3865</spage><epage>3872</epage><pages>3865-3872</pages><issn>0018-9448</issn><eissn>1557-9654</eissn><coden>IETTAW</coden><abstract>The Gilbert-Varshamov (GV) bound states that the maximum size A 2 (n, d) of a binary code of length n and minimum distance d satisfies A 2 (n, d)ges2 n /V(n, d-1) where V(n, d)=Sigma i=0 d ( i n ) stands for the volume of a Hamming ball of radius d. 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| subjects | Applied sciences Asymptotic properties Binary codes Binary system Codes Coding, codes Computer Science Double circulant code Exact sciences and technology Gilbert-Varshamov (GV) bound Graph theory H infinity control Hamming distance Information processing Information Theory Information, signal and communications theory Linear code Mathematics Nonlinear systems Nonlinearity random coding Signal and communications theory Stands Supports Telecommunications and information theory Welding |
| title | Asymptotic Improvement of the Gilbert-Varshamov Bound for Linear Codes |
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