Classification of Binary Constant Weight Codes
A binary code C ⊆ F 2 n with minimum distance at least d and codewords of Hamming weight w is called an (n , d , w ) constant weight code. The maximum size of an (n , d , w ) constant weight code is denoted by A ( n , d , w ), and codes of this size are said to be optimal. In a computer-aided approa...
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| description | A binary code C ⊆ F 2 n with minimum distance at least d and codewords of Hamming weight w is called an (n , d , w ) constant weight code. The maximum size of an (n , d , w ) constant weight code is denoted by A ( n , d , w ), and codes of this size are said to be optimal. In a computer-aided approach, optimal (n , d , w ) constant weight codes are here classified up to equivalence for d =4, n ≤ 12; d =6, n ≤ 14; d =8, n ≤ 17; d =10, n ≤ 20 (with one exception); d =12, n ≤ 23; d =14, n ≤ 26; d =16, n ≤ 28; and d =18, n ≤ 28. Moreover, several new upper bounds on A ( n , d , w ) are obtained, leading among other things to the exact values A (12,4,5)=80, A (15,6,7)=69, A (18,8,7)=33, A (19,8,7)=52, A (19,8,8)=78, and A (20,8,8)=130 . Since A (15,6,6)=70, this gives the first known example of parameters for which A ( n , d , w -1) > A ( n , d , w ) with w ≤ n /2. A scheme based on double counting is developed for validating the classification results. |
| doi_str_mv | 10.1109/TIT.2010.2050922 |
| format | Article |
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The maximum size of an (n , d , w ) constant weight code is denoted by A ( n , d , w ), and codes of this size are said to be optimal. In a computer-aided approach, optimal (n , d , w ) constant weight codes are here classified up to equivalence for d =4, n ≤ 12; d =6, n ≤ 14; d =8, n ≤ 17; d =10, n ≤ 20 (with one exception); d =12, n ≤ 23; d =14, n ≤ 26; d =16, n ≤ 28; and d =18, n ≤ 28. Moreover, several new upper bounds on A ( n , d , w ) are obtained, leading among other things to the exact values A (12,4,5)=80, A (15,6,7)=69, A (18,8,7)=33, A (19,8,7)=52, A (19,8,8)=78, and A (20,8,8)=130 . Since A (15,6,6)=70, this gives the first known example of parameters for which A ( n , d , w -1) > A ( n , d , w ) with w ≤ n /2. A scheme based on double counting is developed for validating the classification results.</description><identifier>ISSN: 0018-9448</identifier><identifier>EISSN: 1557-9654</identifier><identifier>DOI: 10.1109/TIT.2010.2050922</identifier><identifier>CODEN: IETTAW</identifier><language>eng</language><publisher>New York: IEEE</publisher><subject>Binary codes ; Classification ; code equivalence ; Codes ; constant weight code ; Counting ; double counting ; Electronic mail ; Hamming distance ; Hamming weight ; Information theory ; Johnson bound ; Optimization ; Symbols ; Upper bound ; Upper bounds</subject><ispartof>IEEE transactions on information theory, 2010-08, Vol.56 (8), p.3779-3785</ispartof><rights>Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) Aug 2010</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c322t-7c7d444d631642ef86d6285cf871311282643aeaf66c1e02edd259428ff1ee8a3</citedby><cites>FETCH-LOGICAL-c322t-7c7d444d631642ef86d6285cf871311282643aeaf66c1e02edd259428ff1ee8a3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://ieeexplore.ieee.org/document/5508621$$EHTML$$P50$$Gieee$$H</linktohtml><link.rule.ids>314,780,784,796,27924,27925,54758</link.rule.ids><linktorsrc>$$Uhttps://ieeexplore.ieee.org/document/5508621$$EView_record_in_IEEE$$FView_record_in_$$GIEEE</linktorsrc></links><search><creatorcontrib>Ostergard, P R J</creatorcontrib><title>Classification of Binary Constant Weight Codes</title><title>IEEE transactions on information theory</title><addtitle>TIT</addtitle><description>A binary code C ⊆ F 2 n with minimum distance at least d and codewords of Hamming weight w is called an (n , d , w ) constant weight code. The maximum size of an (n , d , w ) constant weight code is denoted by A ( n , d , w ), and codes of this size are said to be optimal. In a computer-aided approach, optimal (n , d , w ) constant weight codes are here classified up to equivalence for d =4, n ≤ 12; d =6, n ≤ 14; d =8, n ≤ 17; d =10, n ≤ 20 (with one exception); d =12, n ≤ 23; d =14, n ≤ 26; d =16, n ≤ 28; and d =18, n ≤ 28. Moreover, several new upper bounds on A ( n , d , w ) are obtained, leading among other things to the exact values A (12,4,5)=80, A (15,6,7)=69, A (18,8,7)=33, A (19,8,7)=52, A (19,8,8)=78, and A (20,8,8)=130 . Since A (15,6,6)=70, this gives the first known example of parameters for which A ( n , d , w -1) > A ( n , d , w ) with w ≤ n /2. A scheme based on double counting is developed for validating the classification results.</description><subject>Binary codes</subject><subject>Classification</subject><subject>code equivalence</subject><subject>Codes</subject><subject>constant weight code</subject><subject>Counting</subject><subject>double counting</subject><subject>Electronic mail</subject><subject>Hamming distance</subject><subject>Hamming weight</subject><subject>Information theory</subject><subject>Johnson bound</subject><subject>Optimization</subject><subject>Symbols</subject><subject>Upper bound</subject><subject>Upper bounds</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2010</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNpdkL1PwzAQxS0EEqWwI7FELEwJvovtOCNEfFSqxFLEaFnOGVylSYnTgf8eV60YmE5P-r17d4-xa-AFAK_vV4tVgTwp5JLXiCdsBlJWea2kOGUzzkHntRD6nF3EuE5SSMAZK5rOxhh8cHYKQ58NPnsMvR1_smbo42T7Kfug8Pk1Jd1SvGRn3naRro5zzt6fn1bNa758e1k0D8vclYhTXrmqFUK0qgQlkLxWrUItndcVlACoUYnSkvVKOSCO1LYoa4HaeyDStpyzu8Pe7Th87yhOZhOio66zPQ27aDRonZJUmcjbf-R62I19Os5UIkWB1CpB_AC5cYhxJG-2Y9ikLw1ws6_PpPrMvj5zrC9Zbg6WQER_uJRcK4TyF2IfaO8</recordid><startdate>201008</startdate><enddate>201008</enddate><creator>Ostergard, P R J</creator><general>IEEE</general><general>The Institute of Electrical and Electronics Engineers, Inc. (IEEE)</general><scope>97E</scope><scope>RIA</scope><scope>RIE</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></search><sort><creationdate>201008</creationdate><title>Classification of Binary Constant Weight Codes</title><author>Ostergard, P R J</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c322t-7c7d444d631642ef86d6285cf871311282643aeaf66c1e02edd259428ff1ee8a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2010</creationdate><topic>Binary codes</topic><topic>Classification</topic><topic>code equivalence</topic><topic>Codes</topic><topic>constant weight code</topic><topic>Counting</topic><topic>double counting</topic><topic>Electronic mail</topic><topic>Hamming distance</topic><topic>Hamming weight</topic><topic>Information theory</topic><topic>Johnson bound</topic><topic>Optimization</topic><topic>Symbols</topic><topic>Upper bound</topic><topic>Upper bounds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ostergard, P R J</creatorcontrib><collection>IEEE All-Society Periodicals Package (ASPP) 2005-present</collection><collection>IEEE All-Society Periodicals Package (ASPP) 1998-Present</collection><collection>IEEE Electronic Library (IEL)</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><jtitle>IEEE transactions on information theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Ostergard, P R J</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Classification of Binary Constant Weight Codes</atitle><jtitle>IEEE transactions on information theory</jtitle><stitle>TIT</stitle><date>2010-08</date><risdate>2010</risdate><volume>56</volume><issue>8</issue><spage>3779</spage><epage>3785</epage><pages>3779-3785</pages><issn>0018-9448</issn><eissn>1557-9654</eissn><coden>IETTAW</coden><abstract>A binary code C ⊆ F 2 n with minimum distance at least d and codewords of Hamming weight w is called an (n , d , w ) constant weight code. The maximum size of an (n , d , w ) constant weight code is denoted by A ( n , d , w ), and codes of this size are said to be optimal. In a computer-aided approach, optimal (n , d , w ) constant weight codes are here classified up to equivalence for d =4, n ≤ 12; d =6, n ≤ 14; d =8, n ≤ 17; d =10, n ≤ 20 (with one exception); d =12, n ≤ 23; d =14, n ≤ 26; d =16, n ≤ 28; and d =18, n ≤ 28. Moreover, several new upper bounds on A ( n , d , w ) are obtained, leading among other things to the exact values A (12,4,5)=80, A (15,6,7)=69, A (18,8,7)=33, A (19,8,7)=52, A (19,8,8)=78, and A (20,8,8)=130 . Since A (15,6,6)=70, this gives the first known example of parameters for which A ( n , d , w -1) > A ( n , d , w ) with w ≤ n /2. A scheme based on double counting is developed for validating the classification results.</abstract><cop>New York</cop><pub>IEEE</pub><doi>10.1109/TIT.2010.2050922</doi><tpages>7</tpages></addata></record> |
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| subjects | Binary codes Classification code equivalence Codes constant weight code Counting double counting Electronic mail Hamming distance Hamming weight Information theory Johnson bound Optimization Symbols Upper bound Upper bounds |
| title | Classification of Binary Constant Weight Codes |
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