On the algebraic structure of quasi-cyclic codes I : finite fields
A new algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese remainder theorem (CRT), or of the discrete Fourier transform (DFT), that ring can be decomposed into a direct p...
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| Format: | Journal Article |
| Idioma: | English |
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2013
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| Accés en línia: | https://hdl.handle.net/10356/96416 http://hdl.handle.net/10220/9827 |
| _version_ | 1826124425653649408 |
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| author | Ling, San Sole, Patrick |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Ling, San Sole, Patrick |
| author_sort | Ling, San |
| collection | NTU |
| description | A new algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese remainder theorem (CRT), or of the discrete Fourier transform (DFT), that ring can be decomposed into a direct product of fields. That ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the (u+υ|u-υ) and Vandermonde constructions. All binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction. Quinting and septing constructions are introduced. Other results made possible by the ring decomposition are a characterization of self-dual quasi-cyclic codes, and a trace representation that generalizes that of cyclic codes. |
| first_indexed | 2024-10-01T06:20:24Z |
| format | Journal Article |
| id | ntu-10356/96416 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T06:20:24Z |
| publishDate | 2013 |
| record_format | dspace |
| spelling | ntu-10356/964162023-02-28T19:22:53Z On the algebraic structure of quasi-cyclic codes I : finite fields Ling, San Sole, Patrick School of Physical and Mathematical Sciences DRNTU::Engineering::Computer science and engineering::Computing methodologies::Symbolic and algebraic manipulation A new algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese remainder theorem (CRT), or of the discrete Fourier transform (DFT), that ring can be decomposed into a direct product of fields. That ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the (u+υ|u-υ) and Vandermonde constructions. All binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction. Quinting and septing constructions are introduced. Other results made possible by the ring decomposition are a characterization of self-dual quasi-cyclic codes, and a trace representation that generalizes that of cyclic codes. Accepted version 2013-04-18T04:22:32Z 2019-12-06T19:30:22Z 2013-04-18T04:22:32Z 2019-12-06T19:30:22Z 2001 2001 Journal Article Ling, S., & Solé, P. (2001). On the algebraic structure of quasi-cyclic codes I: Finite fields. IEEE Transactions on Information Theory, 47(7), 2751-2760. 0018-9448 https://hdl.handle.net/10356/96416 http://hdl.handle.net/10220/9827 10.1109/18.959257 en IEEE transactions on information theory © 2001 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. The published version is available at: [DOI: http://dx.doi.org/10.1109/18.959257]. application/pdf |
| spellingShingle | DRNTU::Engineering::Computer science and engineering::Computing methodologies::Symbolic and algebraic manipulation Ling, San Sole, Patrick On the algebraic structure of quasi-cyclic codes I : finite fields |
| title | On the algebraic structure of quasi-cyclic codes I : finite fields |
| title_full | On the algebraic structure of quasi-cyclic codes I : finite fields |
| title_fullStr | On the algebraic structure of quasi-cyclic codes I : finite fields |
| title_full_unstemmed | On the algebraic structure of quasi-cyclic codes I : finite fields |
| title_short | On the algebraic structure of quasi-cyclic codes I : finite fields |
| title_sort | on the algebraic structure of quasi cyclic codes i finite fields |
| topic | DRNTU::Engineering::Computer science and engineering::Computing methodologies::Symbolic and algebraic manipulation |
| url | https://hdl.handle.net/10356/96416 http://hdl.handle.net/10220/9827 |
| work_keys_str_mv | AT lingsan onthealgebraicstructureofquasicycliccodesifinitefields AT solepatrick onthealgebraicstructureofquasicycliccodesifinitefields |