The normalized second moment of the binary lattice determined by a convolutional code
Calculates the per-dimension mean squared error /spl mu/(S) of the two-state convolutional code C with generator matrix /spl lsqb/1,1+D/spl rsqb/, for the symmetric binary source S=(0,1), and for the uniform source S=/spl lcub/0,1/spl rcub/. When S=(0,1), the quantity /spl mu/(S) is the second momen...
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Veröffentlicht in: | IEEE transactions on information theory 1994-01, Vol.40 (1), p.166-174 |
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description | Calculates the per-dimension mean squared error /spl mu/(S) of the two-state convolutional code C with generator matrix /spl lsqb/1,1+D/spl rsqb/, for the symmetric binary source S=(0,1), and for the uniform source S=/spl lcub/0,1/spl rcub/. When S=(0,1), the quantity /spl mu/(S) is the second moment of the coset weight distribution, which gives the expected Hamming distance of a random binary sequence from the code. When S=/spl lcub/0,1/spl rcub/, the quantity /spl mu/(S) is the second moment of the Voronoi region of the module 2 binary lattice determined by C. The key observation is that a convolutional code with 2/sup /spl upsi// states gives 2/sup /spl upsi// approximations to a given source sequence, and these approximations do not differ very much. It is possible to calculate the steady state distribution for the differences in these path metrics, and hence, the second moment. The authors only give details for the convolutional code /spl lsqb/1,1+D/spl rsqb/, but the method applies to arbitrary codes. They also define the covering radius of a convolutional code, and calculate this quantity for the code /spl lsqb/1,1+D/spl rsqb/.< > |
doi_str_mv | 10.1109/18.272475 |
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When S=(0,1), the quantity /spl mu/(S) is the second moment of the coset weight distribution, which gives the expected Hamming distance of a random binary sequence from the code. When S=/spl lcub/0,1/spl rcub/, the quantity /spl mu/(S) is the second moment of the Voronoi region of the module 2 binary lattice determined by C. The key observation is that a convolutional code with 2/sup /spl upsi// states gives 2/sup /spl upsi// approximations to a given source sequence, and these approximations do not differ very much. It is possible to calculate the steady state distribution for the differences in these path metrics, and hence, the second moment. The authors only give details for the convolutional code /spl lsqb/1,1+D/spl rsqb/, but the method applies to arbitrary codes. They also define the covering radius of a convolutional code, and calculate this quantity for the code /spl lsqb/1,1+D/spl rsqb/.< ></description><identifier>ISSN: 0018-9448</identifier><identifier>EISSN: 1557-9654</identifier><identifier>DOI: 10.1109/18.272475</identifier><identifier>CODEN: IETTAW</identifier><language>eng</language><publisher>New York, NY: IEEE</publisher><subject>Applied sciences ; Binary sequences ; Coding, codes ; Convolutional codes ; Decoding ; Event detection ; Exact sciences and technology ; Hafnium ; Hamming distance ; Information theory ; Information, signal and communications theory ; Lattices ; Quantization ; Signal and communications theory ; Symmetric matrices ; Telecommunications and information theory</subject><ispartof>IEEE transactions on information theory, 1994-01, Vol.40 (1), p.166-174</ispartof><rights>1994 INIST-CNRS</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c306t-335686fa943a2dd76f730a28076113c751ab19d8c298053a62a9f30b5cdc53193</citedby><cites>FETCH-LOGICAL-c306t-335686fa943a2dd76f730a28076113c751ab19d8c298053a62a9f30b5cdc53193</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://ieeexplore.ieee.org/document/272475$$EHTML$$P50$$Gieee$$H</linktohtml><link.rule.ids>314,776,780,792,4010,27900,27901,27902,54733</link.rule.ids><linktorsrc>$$Uhttps://ieeexplore.ieee.org/document/272475$$EView_record_in_IEEE$$FView_record_in_$$GIEEE</linktorsrc><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=4035622$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Calderbank, A.R.</creatorcontrib><creatorcontrib>Fishburn, P.C.</creatorcontrib><title>The normalized second moment of the binary lattice determined by a convolutional code</title><title>IEEE transactions on information theory</title><addtitle>TIT</addtitle><description>Calculates the per-dimension mean squared error /spl mu/(S) of the two-state convolutional code C with generator matrix /spl lsqb/1,1+D/spl rsqb/, for the symmetric binary source S=(0,1), and for the uniform source S=/spl lcub/0,1/spl rcub/. When S=(0,1), the quantity /spl mu/(S) is the second moment of the coset weight distribution, which gives the expected Hamming distance of a random binary sequence from the code. When S=/spl lcub/0,1/spl rcub/, the quantity /spl mu/(S) is the second moment of the Voronoi region of the module 2 binary lattice determined by C. The key observation is that a convolutional code with 2/sup /spl upsi// states gives 2/sup /spl upsi// approximations to a given source sequence, and these approximations do not differ very much. It is possible to calculate the steady state distribution for the differences in these path metrics, and hence, the second moment. The authors only give details for the convolutional code /spl lsqb/1,1+D/spl rsqb/, but the method applies to arbitrary codes. They also define the covering radius of a convolutional code, and calculate this quantity for the code /spl lsqb/1,1+D/spl rsqb/.< ></description><subject>Applied sciences</subject><subject>Binary sequences</subject><subject>Coding, codes</subject><subject>Convolutional codes</subject><subject>Decoding</subject><subject>Event detection</subject><subject>Exact sciences and technology</subject><subject>Hafnium</subject><subject>Hamming distance</subject><subject>Information theory</subject><subject>Information, signal and communications theory</subject><subject>Lattices</subject><subject>Quantization</subject><subject>Signal and communications theory</subject><subject>Symmetric matrices</subject><subject>Telecommunications and information theory</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1994</creationdate><recordtype>article</recordtype><recordid>eNo9kEtLAzEUhYMoWKsLt66yEMHF1Dwmr6WILyi4addDJslgZCapSSrUX2-kpavL4XzncO8F4BqjBcZIPWC5IIK0gp2AGWZMNIqz9hTMEMKyUW0rz8FFzl9VtgyTGVivPh0MMU169L_OwuxMDBZOcXKhwDjAUv3eB512cNSleOOgdcWlyYeK9zuoYU38xHFbfAx6rMq6S3A26DG7q8Ocg_XL8-rprVl-vL4_PS4bQxEvDaWMSz5o1VJNrBV8EBRpIpHgGFMjGNY9VlYaoiRiVHOi1UBRz4w1jGJF5-Bu37tJ8Xvrcukmn40bRx1c3OaOSC4YlaKC93vQpJhzckO3SX6qR3UYdf-P67Ds9o-r7O2hVGejxyHpYHw-BlpUtyakYjd7zDvnju6h4w8cKnRa</recordid><startdate>199401</startdate><enddate>199401</enddate><creator>Calderbank, A.R.</creator><creator>Fishburn, P.C.</creator><general>IEEE</general><general>Institute of Electrical and Electronics Engineers</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>199401</creationdate><title>The normalized second moment of the binary lattice determined by a convolutional code</title><author>Calderbank, A.R. ; Fishburn, P.C.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c306t-335686fa943a2dd76f730a28076113c751ab19d8c298053a62a9f30b5cdc53193</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1994</creationdate><topic>Applied sciences</topic><topic>Binary sequences</topic><topic>Coding, codes</topic><topic>Convolutional codes</topic><topic>Decoding</topic><topic>Event detection</topic><topic>Exact sciences and technology</topic><topic>Hafnium</topic><topic>Hamming distance</topic><topic>Information theory</topic><topic>Information, signal and communications theory</topic><topic>Lattices</topic><topic>Quantization</topic><topic>Signal and communications theory</topic><topic>Symmetric matrices</topic><topic>Telecommunications and information theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Calderbank, A.R.</creatorcontrib><creatorcontrib>Fishburn, P.C.</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems 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><jtitle>IEEE transactions on information theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Calderbank, A.R.</au><au>Fishburn, P.C.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The normalized second moment of the binary lattice determined by a convolutional code</atitle><jtitle>IEEE transactions on information theory</jtitle><stitle>TIT</stitle><date>1994-01</date><risdate>1994</risdate><volume>40</volume><issue>1</issue><spage>166</spage><epage>174</epage><pages>166-174</pages><issn>0018-9448</issn><eissn>1557-9654</eissn><coden>IETTAW</coden><abstract>Calculates the per-dimension mean squared error /spl mu/(S) of the two-state convolutional code C with generator matrix /spl lsqb/1,1+D/spl rsqb/, for the symmetric binary source S=(0,1), and for the uniform source S=/spl lcub/0,1/spl rcub/. When S=(0,1), the quantity /spl mu/(S) is the second moment of the coset weight distribution, which gives the expected Hamming distance of a random binary sequence from the code. When S=/spl lcub/0,1/spl rcub/, the quantity /spl mu/(S) is the second moment of the Voronoi region of the module 2 binary lattice determined by C. The key observation is that a convolutional code with 2/sup /spl upsi// states gives 2/sup /spl upsi// approximations to a given source sequence, and these approximations do not differ very much. It is possible to calculate the steady state distribution for the differences in these path metrics, and hence, the second moment. The authors only give details for the convolutional code /spl lsqb/1,1+D/spl rsqb/, but the method applies to arbitrary codes. They also define the covering radius of a convolutional code, and calculate this quantity for the code /spl lsqb/1,1+D/spl rsqb/.< ></abstract><cop>New York, NY</cop><pub>IEEE</pub><doi>10.1109/18.272475</doi><tpages>9</tpages></addata></record> |
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subjects | Applied sciences Binary sequences Coding, codes Convolutional codes Decoding Event detection Exact sciences and technology Hafnium Hamming distance Information theory Information, signal and communications theory Lattices Quantization Signal and communications theory Symmetric matrices Telecommunications and information theory |
title | The normalized second moment of the binary lattice determined by a convolutional code |
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