{"title":"若干线性码的停止集分布","authors":"Yong Jiang, Shutao Xia, Fang-Wei Fu","doi":"10.1109/ITW2.2006.323751","DOIUrl":null,"url":null,"abstract":"In this paper, the stopping set distributions (SSD) of some well-known binary linear codes are determined by using finite geometry theory. Similar to the weight distribution of a binary linear code, the SSD {Ti(H)}n i=0 enumerates the number of stopping sets with size i of a linear code with parity-check matrix H. First, we deal with the simplex codes and Hamming codes. With parity-check matrix formed by all the weight 3 codewords of the Hamming code, the SSD of the simplex code is completely determined with explicit formula. With parity-check matrix formed by all the nonzero codewords of the simplex code, the SSD of the Hamming code is completely determined with two recursive equations. Then, the first order Reed-Muller codes and the extended Hamming codes are discussed. With parity-check matrix formed by all the weight 4 codewords of the extended Hamming code, the SSD of the first order Reed-Muller code is completely determined with explicit formula. With parity-check matrix formed by all the minimum codewords of the first order Reed-Muller code, the SSD of the extended Hamming code is completely determined with two recursive equations","PeriodicalId":299513,"journal":{"name":"2006 IEEE Information Theory Workshop - ITW '06 Chengdu","volume":"284 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2006-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"Stopping Set Distributions of Some Linear Codes\",\"authors\":\"Yong Jiang, Shutao Xia, Fang-Wei Fu\",\"doi\":\"10.1109/ITW2.2006.323751\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, the stopping set distributions (SSD) of some well-known binary linear codes are determined by using finite geometry theory. Similar to the weight distribution of a binary linear code, the SSD {Ti(H)}n i=0 enumerates the number of stopping sets with size i of a linear code with parity-check matrix H. First, we deal with the simplex codes and Hamming codes. With parity-check matrix formed by all the weight 3 codewords of the Hamming code, the SSD of the simplex code is completely determined with explicit formula. With parity-check matrix formed by all the nonzero codewords of the simplex code, the SSD of the Hamming code is completely determined with two recursive equations. Then, the first order Reed-Muller codes and the extended Hamming codes are discussed. With parity-check matrix formed by all the weight 4 codewords of the extended Hamming code, the SSD of the first order Reed-Muller code is completely determined with explicit formula. With parity-check matrix formed by all the minimum codewords of the first order Reed-Muller code, the SSD of the extended Hamming code is completely determined with two recursive equations\",\"PeriodicalId\":299513,\"journal\":{\"name\":\"2006 IEEE Information Theory Workshop - ITW '06 Chengdu\",\"volume\":\"284 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2006-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2006 IEEE Information Theory Workshop - ITW '06 Chengdu\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/ITW2.2006.323751\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2006 IEEE Information Theory Workshop - ITW '06 Chengdu","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/ITW2.2006.323751","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
In this paper, the stopping set distributions (SSD) of some well-known binary linear codes are determined by using finite geometry theory. Similar to the weight distribution of a binary linear code, the SSD {Ti(H)}n i=0 enumerates the number of stopping sets with size i of a linear code with parity-check matrix H. First, we deal with the simplex codes and Hamming codes. With parity-check matrix formed by all the weight 3 codewords of the Hamming code, the SSD of the simplex code is completely determined with explicit formula. With parity-check matrix formed by all the nonzero codewords of the simplex code, the SSD of the Hamming code is completely determined with two recursive equations. Then, the first order Reed-Muller codes and the extended Hamming codes are discussed. With parity-check matrix formed by all the weight 4 codewords of the extended Hamming code, the SSD of the first order Reed-Muller code is completely determined with explicit formula. With parity-check matrix formed by all the minimum codewords of the first order Reed-Muller code, the SSD of the extended Hamming code is completely determined with two recursive equations
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