Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order Two
Suppose that p and q are two distinct odd prime numbers with n=pq. In this paper, the uniform representation of general two-prime generalized cyclotomy with order two over ℤn was demonstrated. Based on this general generalized cyclotomy, a type of binary sequences defined over Fl was presented and t...
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| Format: | Article |
| Language: | English |
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Wiley
2020-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2020/6625652 |
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| author | Xia Zhou |
| author_facet | Xia Zhou |
| author_sort | Xia Zhou |
| collection | DOAJ |
| description | Suppose that p and q are two distinct odd prime numbers with n=pq. In this paper, the uniform representation of general two-prime generalized cyclotomy with order two over ℤn was demonstrated. Based on this general generalized cyclotomy, a type of binary sequences defined over Fl was presented and their minimal polynomials and linear complexities were derived, where l=rs with a prime number r and gcdl,n=1. The results have indicated that the linear complexities of these sequences are high without any special requirements on the prime numbers. Furthermore, we employed these sequences to obtain a few cyclic codes over Fl with length n and developed the lower bounds of the minimum distances of many cyclic codes. It is important to stress that some cyclic codes in this paper are optimal. |
| format | Article |
| id | doaj-art-4c0303ff66424a2d96d97a39bc697e74 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-4c0303ff66424a2d96d97a39bc697e742025-02-03T01:25:46ZengWileyJournal of Mathematics2314-46292314-47852020-01-01202010.1155/2020/66256526625652Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order TwoXia Zhou0School of Economics and Management, Nanjing Institute of Technology, Nanjing 211167, ChinaSuppose that p and q are two distinct odd prime numbers with n=pq. In this paper, the uniform representation of general two-prime generalized cyclotomy with order two over ℤn was demonstrated. Based on this general generalized cyclotomy, a type of binary sequences defined over Fl was presented and their minimal polynomials and linear complexities were derived, where l=rs with a prime number r and gcdl,n=1. The results have indicated that the linear complexities of these sequences are high without any special requirements on the prime numbers. Furthermore, we employed these sequences to obtain a few cyclic codes over Fl with length n and developed the lower bounds of the minimum distances of many cyclic codes. It is important to stress that some cyclic codes in this paper are optimal.http://dx.doi.org/10.1155/2020/6625652 |
| spellingShingle | Xia Zhou Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order Two Journal of Mathematics |
| title | Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order Two |
| title_full | Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order Two |
| title_fullStr | Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order Two |
| title_full_unstemmed | Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order Two |
| title_short | Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order Two |
| title_sort | cyclic codes via the general two prime generalized cyclotomic sequence of order two |
| url | http://dx.doi.org/10.1155/2020/6625652 |
| work_keys_str_mv | AT xiazhou cycliccodesviathegeneraltwoprimegeneralizedcyclotomicsequenceofordertwo |