{"title":"一阶Reed-Muller码的线性复杂度软判决表译码","authors":"I. Dumer, G. Kabatiansky, C. Tavernier","doi":"10.1109/ISIT.2007.4557410","DOIUrl":null,"url":null,"abstract":"Soft-decision decoding on a memoryless channel is considered for the first-order Reed-Muller codes RM(1,m) of length 2m. We assume that different positions j of the received binary vector y can be corrupted by the errors of varying weight Wj. The generalized Hamming distance between vector y and any binary vector c is then defined as the sum of weighted differences Wj|yj - Cj| taken over all n positions. We obtain a tight upper bound Lt on the number of codewords located within generalized Hamming distance T from vector y, and design a decoding algorithm that outputs this list of codewords with complexity O(n In2 Lt)- In particular, all possible error weights wj equal 1 if this combinatorial model is applied to a binary symmetric channel. In this case, the well known Green algorithm performs full maximum likelihood decoding of RM(1,m) and requires O(n ln2 n) bit operations, whereas the Litsyn-Shekhovtsov algorithm operates within the bounded-distance decoding radius n/4 - 1 with linear complexity O(n). We close the performance-complexity gap between the two algorithms. Namely, for any fixed epsi isin (0, frac12), our algorithm outputs the complete list of codewords within the decoding radius n(frac12 - epsi) with linear complexity of order n ln2 epsi.","PeriodicalId":193467,"journal":{"name":"2007 IEEE International Symposium on Information Theory","volume":"7 4 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2007-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Soft-Decision List Decoding with Linear Complexity for the First-Order Reed-Muller Codes\",\"authors\":\"I. Dumer, G. Kabatiansky, C. Tavernier\",\"doi\":\"10.1109/ISIT.2007.4557410\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Soft-decision decoding on a memoryless channel is considered for the first-order Reed-Muller codes RM(1,m) of length 2m. We assume that different positions j of the received binary vector y can be corrupted by the errors of varying weight Wj. The generalized Hamming distance between vector y and any binary vector c is then defined as the sum of weighted differences Wj|yj - Cj| taken over all n positions. We obtain a tight upper bound Lt on the number of codewords located within generalized Hamming distance T from vector y, and design a decoding algorithm that outputs this list of codewords with complexity O(n In2 Lt)- In particular, all possible error weights wj equal 1 if this combinatorial model is applied to a binary symmetric channel. In this case, the well known Green algorithm performs full maximum likelihood decoding of RM(1,m) and requires O(n ln2 n) bit operations, whereas the Litsyn-Shekhovtsov algorithm operates within the bounded-distance decoding radius n/4 - 1 with linear complexity O(n). We close the performance-complexity gap between the two algorithms. Namely, for any fixed epsi isin (0, frac12), our algorithm outputs the complete list of codewords within the decoding radius n(frac12 - epsi) with linear complexity of order n ln2 epsi.\",\"PeriodicalId\":193467,\"journal\":{\"name\":\"2007 IEEE International Symposium on Information Theory\",\"volume\":\"7 4 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2007-06-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2007 IEEE International Symposium on Information Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/ISIT.2007.4557410\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2007 IEEE International Symposium on Information Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/ISIT.2007.4557410","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Soft-Decision List Decoding with Linear Complexity for the First-Order Reed-Muller Codes
Soft-decision decoding on a memoryless channel is considered for the first-order Reed-Muller codes RM(1,m) of length 2m. We assume that different positions j of the received binary vector y can be corrupted by the errors of varying weight Wj. The generalized Hamming distance between vector y and any binary vector c is then defined as the sum of weighted differences Wj|yj - Cj| taken over all n positions. We obtain a tight upper bound Lt on the number of codewords located within generalized Hamming distance T from vector y, and design a decoding algorithm that outputs this list of codewords with complexity O(n In2 Lt)- In particular, all possible error weights wj equal 1 if this combinatorial model is applied to a binary symmetric channel. In this case, the well known Green algorithm performs full maximum likelihood decoding of RM(1,m) and requires O(n ln2 n) bit operations, whereas the Litsyn-Shekhovtsov algorithm operates within the bounded-distance decoding radius n/4 - 1 with linear complexity O(n). We close the performance-complexity gap between the two algorithms. Namely, for any fixed epsi isin (0, frac12), our algorithm outputs the complete list of codewords within the decoding radius n(frac12 - epsi) with linear complexity of order n ln2 epsi.