{"title":"A Geometrical Realisation of Quasi-Cyclic Codes","authors":"Cristina Martínez Ramírez, Alberto Besana","doi":"10.5772/intechopen.88288","DOIUrl":null,"url":null,"abstract":"We study and enumerate cyclic codes which include generalised Reed-Solomon codes as function field codes. This geometrical approach allows to construct longer codes and to get more information on the parameters defining the codes. We provide a closed formula in terms of Stirling numbers for the number of irreducible polynomials and we relate it with other formulas existing in the literature. Further, we study quasi-cyclic codes as orbit codes in the Grassmannian parameterizing constant dimension codes. In addition, we review Horn ’ s algorithm and apply it to construct classical codes by their defining ideals.","PeriodicalId":184595,"journal":{"name":"Probability, Combinatorics and Control","volume":"20 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Probability, Combinatorics and Control","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5772/intechopen.88288","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 3
Abstract
We study and enumerate cyclic codes which include generalised Reed-Solomon codes as function field codes. This geometrical approach allows to construct longer codes and to get more information on the parameters defining the codes. We provide a closed formula in terms of Stirling numbers for the number of irreducible polynomials and we relate it with other formulas existing in the literature. Further, we study quasi-cyclic codes as orbit codes in the Grassmannian parameterizing constant dimension codes. In addition, we review Horn ’ s algorithm and apply it to construct classical codes by their defining ideals.
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