{"title":"渐进速率与设计速率","authors":"C. Méasson, A. Montanari, R. Urbanke","doi":"10.1109/ISIT.2007.4557441","DOIUrl":null,"url":null,"abstract":"The rate of a code is one of its most important parameters. We consider sparse graph codes and ask whether the rate of a random element of an ensemble is typically close to the design rate of the ensemble. For regular LDPC ensembles this question was answered in the affirmative in (Miller and Cohen, 2003). We start by giving an alternative proof of this statement. We then show that essentially the same type of argument applies not only to regular ensembles but also to ensembles that are derived from regular ensembles in the sense that their degree distribution is the result of applying the peeling decoder to a regular code. As an immediate consequence we prove that for regular ensembles the asymptotic MAP EXIT value coincides with the asymptotic BP EXIT value. We then give a systematic construction of ensembles for which rate and design rate differ. To accomplish this, we first show that the duality theorem (Ashikhminet al., 2004) implies that the asymptotic BP EXIT and the MAP EXIT functions are identical for any channel parameter for which the density evolution (DE) equations have a unique fixed point.","PeriodicalId":193467,"journal":{"name":"2007 IEEE International Symposium on Information Theory","volume":"75 4 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2007-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"10","resultStr":"{\"title\":\"Asymptotic Rate versus Design Rate\",\"authors\":\"C. Méasson, A. Montanari, R. Urbanke\",\"doi\":\"10.1109/ISIT.2007.4557441\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The rate of a code is one of its most important parameters. We consider sparse graph codes and ask whether the rate of a random element of an ensemble is typically close to the design rate of the ensemble. For regular LDPC ensembles this question was answered in the affirmative in (Miller and Cohen, 2003). We start by giving an alternative proof of this statement. We then show that essentially the same type of argument applies not only to regular ensembles but also to ensembles that are derived from regular ensembles in the sense that their degree distribution is the result of applying the peeling decoder to a regular code. As an immediate consequence we prove that for regular ensembles the asymptotic MAP EXIT value coincides with the asymptotic BP EXIT value. We then give a systematic construction of ensembles for which rate and design rate differ. To accomplish this, we first show that the duality theorem (Ashikhminet al., 2004) implies that the asymptotic BP EXIT and the MAP EXIT functions are identical for any channel parameter for which the density evolution (DE) equations have a unique fixed point.\",\"PeriodicalId\":193467,\"journal\":{\"name\":\"2007 IEEE International Symposium on Information Theory\",\"volume\":\"75 4 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2007-06-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"10\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2007 IEEE International Symposium on Information Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/ISIT.2007.4557441\",\"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.2007.4557441","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 10
摘要
码率是其最重要的参数之一。我们考虑稀疏图码,并询问集成的随机元素的速率是否通常接近集成的设计速率。对于常规的LDPC组合,这个问题在(Miller and Cohen, 2003)中得到了肯定的回答。我们首先给出这个命题的另一种证明。然后我们表明,本质上相同类型的参数不仅适用于规则集成,也适用于从规则集成派生的集成,因为它们的度分布是将剥离解码器应用于规则代码的结果。作为一个直接的结果,我们证明了对于正则系综,渐近的MAP EXIT值与渐近的BP EXIT值重合。然后,我们给出了速率和设计速率不同的集成系统的系统结构。为了实现这一点,我们首先证明对偶定理(Ashikhminet al., 2004)意味着对于密度演化(DE)方程具有唯一不动点的任何信道参数,渐近BP EXIT和MAP EXIT函数是相同的。
The rate of a code is one of its most important parameters. We consider sparse graph codes and ask whether the rate of a random element of an ensemble is typically close to the design rate of the ensemble. For regular LDPC ensembles this question was answered in the affirmative in (Miller and Cohen, 2003). We start by giving an alternative proof of this statement. We then show that essentially the same type of argument applies not only to regular ensembles but also to ensembles that are derived from regular ensembles in the sense that their degree distribution is the result of applying the peeling decoder to a regular code. As an immediate consequence we prove that for regular ensembles the asymptotic MAP EXIT value coincides with the asymptotic BP EXIT value. We then give a systematic construction of ensembles for which rate and design rate differ. To accomplish this, we first show that the duality theorem (Ashikhminet al., 2004) implies that the asymptotic BP EXIT and the MAP EXIT functions are identical for any channel parameter for which the density evolution (DE) equations have a unique fixed point.