Constructions and decoding of a sequence of BCH codes
Autor(a) principal: | |
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Data de Publicação: | 2012 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://hdl.handle.net/11449/227210 |
Resumo: | The BCH code C (respectively, C) of length n over a local ring Z pk (respectively, ℤp) is an ideal in the ring (Equation Presented) (respectively, (Equation Presented) which is generated by a monic polynomial that divides Xn - 1. Shankar [12] has shown that the roots of Xn - 1 are the unit elements of a suitable Galois ring extension GR(pk,s) (respectively, Galois field extension GF(p, s)) of the ring ℤpk (respectively, ℤp), where s is the degree of basic irreducible polynomial f(X) ∈ ℤpk [X]. In this study we assume that for st = bi, where 6 is prime and t is a non negative integer such that 0 ≤ i ≤ t, there exist corresponding chain of Galois ring extensions GR(pk, s,) (respectively, a chain of Galois field extensions GF(p, s,)) of ℤpk (respectively, ℤp), there are two situations; st = bi for i = 2 or st = bi for i ≥ 2. Consequently, the case is alike [12] and we obtain a sequence of BCH codes C0,C1, ···, Ct-1, C over ℤpk and C′0,C′1,···, C′t-1,C′ over ℤp with lengths n 0,n1,···, nt-1,n t. In second phase we extend the Modified Berlekamp-Massey Algorithm for the chain of Galois rings in such a way that the error will be corrected of the sequence of codewords from the sequence of BCH codes C0,C 1, ···, Ct-1,C. © Global Publishing Company. |
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Constructions and decoding of a sequence of BCH codesBCH codeDecodingEncodingGalois fieldGalois ringModified Berlekamp-Massey AlgorithmThe BCH code C (respectively, C) of length n over a local ring Z pk (respectively, ℤp) is an ideal in the ring (Equation Presented) (respectively, (Equation Presented) which is generated by a monic polynomial that divides Xn - 1. Shankar [12] has shown that the roots of Xn - 1 are the unit elements of a suitable Galois ring extension GR(pk,s) (respectively, Galois field extension GF(p, s)) of the ring ℤpk (respectively, ℤp), where s is the degree of basic irreducible polynomial f(X) ∈ ℤpk [X]. In this study we assume that for st = bi, where 6 is prime and t is a non negative integer such that 0 ≤ i ≤ t, there exist corresponding chain of Galois ring extensions GR(pk, s,) (respectively, a chain of Galois field extensions GF(p, s,)) of ℤpk (respectively, ℤp), there are two situations; st = bi for i = 2 or st = bi for i ≥ 2. Consequently, the case is alike [12] and we obtain a sequence of BCH codes C0,C1, ···, Ct-1, C over ℤpk and C′0,C′1,···, C′t-1,C′ over ℤp with lengths n 0,n1,···, nt-1,n t. In second phase we extend the Modified Berlekamp-Massey Algorithm for the chain of Galois rings in such a way that the error will be corrected of the sequence of codewords from the sequence of BCH codes C0,C 1, ···, Ct-1,C. © Global Publishing Company.Department of Mathematics, Quaid-i-Azam University, IslamabadDepartment of Mathematics, São Paulo State University, São José do Rio Preto - SPDepartment of Mathematics, São Paulo State University, São José do Rio Preto - SPUniversidade Estadual Paulista (UNESP)Shah, TaxiqQamar, AttiqDe Andrade, Antonio Aparecido [UNESP]2022-04-29T07:12:08Z2022-04-29T07:12:08Z2012-01-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article234-250Mathematical Sciences Research Journal, v. 16, n. 9, p. 234-250, 2012.1537-5978http://hdl.handle.net/11449/2272102-s2.0-84884528679Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengMathematical Sciences Research Journalinfo:eu-repo/semantics/openAccess2022-04-29T07:12:08Zoai:repositorio.unesp.br:11449/227210Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T16:53:55.986779Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
Constructions and decoding of a sequence of BCH codes |
title |
Constructions and decoding of a sequence of BCH codes |
spellingShingle |
Constructions and decoding of a sequence of BCH codes Shah, Taxiq BCH code Decoding Encoding Galois field Galois ring Modified Berlekamp-Massey Algorithm |
title_short |
Constructions and decoding of a sequence of BCH codes |
title_full |
Constructions and decoding of a sequence of BCH codes |
title_fullStr |
Constructions and decoding of a sequence of BCH codes |
title_full_unstemmed |
Constructions and decoding of a sequence of BCH codes |
title_sort |
Constructions and decoding of a sequence of BCH codes |
author |
Shah, Taxiq |
author_facet |
Shah, Taxiq Qamar, Attiq De Andrade, Antonio Aparecido [UNESP] |
author_role |
author |
author2 |
Qamar, Attiq De Andrade, Antonio Aparecido [UNESP] |
author2_role |
author author |
dc.contributor.none.fl_str_mv |
Universidade Estadual Paulista (UNESP) |
dc.contributor.author.fl_str_mv |
Shah, Taxiq Qamar, Attiq De Andrade, Antonio Aparecido [UNESP] |
dc.subject.por.fl_str_mv |
BCH code Decoding Encoding Galois field Galois ring Modified Berlekamp-Massey Algorithm |
topic |
BCH code Decoding Encoding Galois field Galois ring Modified Berlekamp-Massey Algorithm |
description |
The BCH code C (respectively, C) of length n over a local ring Z pk (respectively, ℤp) is an ideal in the ring (Equation Presented) (respectively, (Equation Presented) which is generated by a monic polynomial that divides Xn - 1. Shankar [12] has shown that the roots of Xn - 1 are the unit elements of a suitable Galois ring extension GR(pk,s) (respectively, Galois field extension GF(p, s)) of the ring ℤpk (respectively, ℤp), where s is the degree of basic irreducible polynomial f(X) ∈ ℤpk [X]. In this study we assume that for st = bi, where 6 is prime and t is a non negative integer such that 0 ≤ i ≤ t, there exist corresponding chain of Galois ring extensions GR(pk, s,) (respectively, a chain of Galois field extensions GF(p, s,)) of ℤpk (respectively, ℤp), there are two situations; st = bi for i = 2 or st = bi for i ≥ 2. Consequently, the case is alike [12] and we obtain a sequence of BCH codes C0,C1, ···, Ct-1, C over ℤpk and C′0,C′1,···, C′t-1,C′ over ℤp with lengths n 0,n1,···, nt-1,n t. In second phase we extend the Modified Berlekamp-Massey Algorithm for the chain of Galois rings in such a way that the error will be corrected of the sequence of codewords from the sequence of BCH codes C0,C 1, ···, Ct-1,C. © Global Publishing Company. |
publishDate |
2012 |
dc.date.none.fl_str_mv |
2012-01-01 2022-04-29T07:12:08Z 2022-04-29T07:12:08Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
Mathematical Sciences Research Journal, v. 16, n. 9, p. 234-250, 2012. 1537-5978 http://hdl.handle.net/11449/227210 2-s2.0-84884528679 |
identifier_str_mv |
Mathematical Sciences Research Journal, v. 16, n. 9, p. 234-250, 2012. 1537-5978 2-s2.0-84884528679 |
url |
http://hdl.handle.net/11449/227210 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Mathematical Sciences Research Journal |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
234-250 |
dc.source.none.fl_str_mv |
Scopus reponame:Repositório Institucional da UNESP instname:Universidade Estadual Paulista (UNESP) instacron:UNESP |
instname_str |
Universidade Estadual Paulista (UNESP) |
instacron_str |
UNESP |
institution |
UNESP |
reponame_str |
Repositório Institucional da UNESP |
collection |
Repositório Institucional da UNESP |
repository.name.fl_str_mv |
Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP) |
repository.mail.fl_str_mv |
|
_version_ |
1808128718593327104 |