{"title":"Hardness of Successive-Cancellation Decoding of Linear Codes","authors":"Arman Fazeli, A. Vardy, Hanwen Yao","doi":"10.1109/ISIT44484.2020.9174469","DOIUrl":null,"url":null,"abstract":"Successive-cancellation decoding has gained much renewed interest since the advent of polar coding a decade ago. For polar codes, successive-cancellation decoding can be accomplished in time O(n log n). However, the complexity of successive-cancellation decoding for other families of codes remains largely unexplored. Herein, we prove that successive-cancellation decoding of general binary linear codes is NP-hard. In order to establish this result, we reduce from maximum-likelihood decoding of linear codes, a well-known NP-complete problem. Unlike maximum-likelihood decoding, however, the successive-cancellation decoding problem depends on the choice of a generator matrix. Thus we further strengthen our result by showing that there exist codes for which successive-cancellation decoding remains hard for every possible choice of the generator matrix. On the other hand, we also observe that polynomial-time successive-cancellation decoding can be extended from polar codes to many other linear codes. Finally, we show that every binary linear code can be encoded as a polar code with dynamically frozen bits. This approach makes it possible to use list-decoding of polar codes to approximate the maximum-likelihood decoding performance of arbitrary codes.","PeriodicalId":159311,"journal":{"name":"2020 IEEE International Symposium on Information Theory (ISIT)","volume":"57 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"2020 IEEE International Symposium on Information Theory (ISIT)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/ISIT44484.2020.9174469","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
Successive-cancellation decoding has gained much renewed interest since the advent of polar coding a decade ago. For polar codes, successive-cancellation decoding can be accomplished in time O(n log n). However, the complexity of successive-cancellation decoding for other families of codes remains largely unexplored. Herein, we prove that successive-cancellation decoding of general binary linear codes is NP-hard. In order to establish this result, we reduce from maximum-likelihood decoding of linear codes, a well-known NP-complete problem. Unlike maximum-likelihood decoding, however, the successive-cancellation decoding problem depends on the choice of a generator matrix. Thus we further strengthen our result by showing that there exist codes for which successive-cancellation decoding remains hard for every possible choice of the generator matrix. On the other hand, we also observe that polynomial-time successive-cancellation decoding can be extended from polar codes to many other linear codes. Finally, we show that every binary linear code can be encoded as a polar code with dynamically frozen bits. This approach makes it possible to use list-decoding of polar codes to approximate the maximum-likelihood decoding performance of arbitrary codes.