Shortening Array Codes and the Perfect 1-Factorization Conjecture
The existence of a perfect 1-factorization of the complete graph with n nodes, namely, K n , for arbitrary even number n, is a 40-year-old open problem in graph theory. So far, two infinite families of perfect 1-factorizations have been shown to exist, namely, the factorizations of Kp+1 and K 2 p ,...
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| Veröffentlicht in: | IEEE transactions on information theory 2009-02, Vol.55 (2), p.507-513 |
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| description | The existence of a perfect 1-factorization of the complete graph with n nodes, namely, K n , for arbitrary even number n, is a 40-year-old open problem in graph theory. So far, two infinite families of perfect 1-factorizations have been shown to exist, namely, the factorizations of Kp+1 and K 2 p , where p is an arbitrary prime number (p > 2) . It was shown in previous work that finding a perfect 1 -factorization of K n is related to a problem in coding, specifically, it can be reduced to constructing an MDS (Minimum Distance Separable), lowest density array code. In this paper, a new method for shortening arbitrary array codes is introduced. It is then used to derive the K p+1 family of perfect 1 -factorization from the K 2p family. Namely, techniques from coding theory are used to prove a new result in graph theory-that the two factorization families are related. |
| doi_str_mv | 10.1109/TIT.2008.2009850 |
| format | Article |
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Namely, techniques from coding theory are used to prove a new result in graph theory-that the two factorization families are related.</description><subject>1 -factorization</subject><subject>Applied sciences</subject><subject>Array codes</subject><subject>Arrays</subject><subject>Codes</subject><subject>Coding</subject><subject>Coding, codes</subject><subject>Construction specifications</subject><subject>Data encryption</subject><subject>Decoding</subject><subject>Density</subject><subject>Error correction codes</subject><subject>error-correcting codes</subject><subject>Exact sciences and technology</subject><subject>Factorization</subject><subject>Graph theory</subject><subject>Graphs</subject><subject>Information theory</subject><subject>Information, signal and communications theory</subject><subject>perfect 1 -factorization</subject><subject>Prime numbers</subject><subject>Signal and communications theory</subject><subject>Telecommunications and information theory</subject><subject>Theory</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2009</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNp9kE1LAzEQhoMoWKt3wcsiKF62ZpJsPo6lWC0UFKznJZtk7ZZ2o8nuof56U1p68OBlhpl55mXmRega8AgAq8fFbDEiGMtdULLAJ2gARSFyxQt2igYYg8wVY_IcXcS4SiUrgAzQ-H3pQ-fapv3MxiHobTbx1sVMtzbrli57c6F2pssgn2rT-dD86K7xbaLaVer3wV2is1qvo7s65CH6mD4tJi_5_PV5NhnPc8MI73JSWWsVpxWlUoER1nBWE0cxmDpdYy2nlnOmrbTGiUIDVApjXCmFa61kTYfofq_7Ffx372JXbppo3HqtW-f7WFLGJeGKJvDhXxAwBaI45zKht3_Qle9Dm94oQRWKCsCQILyHTPAxBleXX6HZ6LBNSuXO-zJ5X-68Lw_ep5W7g66ORq_roFvTxOMeASYIUJG4mz3XOOeOYyaE4EzQX3y4iuQ</recordid><startdate>20090201</startdate><enddate>20090201</enddate><creator>Bohossian, V.</creator><creator>Bruck, J.</creator><general>IEEE</general><general>Institute of Electrical and Electronics Engineers</general><general>The Institute of Electrical and Electronics Engineers, Inc. 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| subjects | 1 -factorization Applied sciences Array codes Arrays Codes Coding Coding, codes Construction specifications Data encryption Decoding Density Error correction codes error-correcting codes Exact sciences and technology Factorization Graph theory Graphs Information theory Information, signal and communications theory perfect 1 -factorization Prime numbers Signal and communications theory Telecommunications and information theory Theory |
| title | Shortening Array Codes and the Perfect 1-Factorization Conjecture |
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