Matrix-Product Codes over Commutative Rings and Constructions Arising from σ,δ-Codes
A well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ring R. A sufficient condition is given, as well, for such a bound to be sharp. It is also shown that an MPC is...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/5521067 |
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| author | Mhammed Boulagouaz Abdulaziz Deajim |
| author_facet | Mhammed Boulagouaz Abdulaziz Deajim |
| author_sort | Mhammed Boulagouaz |
| collection | DOAJ |
| description | A well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ring R. A sufficient condition is given, as well, for such a bound to be sharp. It is also shown that an MPC is free when its input codes are all free, in which case a generator matrix is given. If R is finite, a sufficient condition is provided for the dual of an MPC to be an MPC, a generator matrix for such a dual is given, and characterizations of LCD, self-dual, and self-orthogonal MPCs are presented. Finally, the results of this paper are used along with previous results of the authors to construct novel MPCs arising from σ,δ-codes. Some properties of such constructions are also studied. |
| format | Article |
| id | doaj-art-d6e66441cf374477adfc2fc785e31938 |
| institution | DOAJ |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-d6e66441cf374477adfc2fc785e319382025-08-20T03:23:51ZengWileyJournal of Mathematics2314-46292314-47852021-01-01202110.1155/2021/55210675521067Matrix-Product Codes over Commutative Rings and Constructions Arising from σ,δ-CodesMhammed Boulagouaz0Abdulaziz Deajim1Department of Mathematics, Faculty of Sciences and Technologies, University of Sidi Mohamed Ben Abdellah, B.P. 2202, Fes, MoroccoDepartment of Mathematics, King Khalid University, P.O. Box 9004, Abha, Saudi ArabiaA well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ring R. A sufficient condition is given, as well, for such a bound to be sharp. It is also shown that an MPC is free when its input codes are all free, in which case a generator matrix is given. If R is finite, a sufficient condition is provided for the dual of an MPC to be an MPC, a generator matrix for such a dual is given, and characterizations of LCD, self-dual, and self-orthogonal MPCs are presented. Finally, the results of this paper are used along with previous results of the authors to construct novel MPCs arising from σ,δ-codes. Some properties of such constructions are also studied.http://dx.doi.org/10.1155/2021/5521067 |
| spellingShingle | Mhammed Boulagouaz Abdulaziz Deajim Matrix-Product Codes over Commutative Rings and Constructions Arising from σ,δ-Codes Journal of Mathematics |
| title | Matrix-Product Codes over Commutative Rings and Constructions Arising from σ,δ-Codes |
| title_full | Matrix-Product Codes over Commutative Rings and Constructions Arising from σ,δ-Codes |
| title_fullStr | Matrix-Product Codes over Commutative Rings and Constructions Arising from σ,δ-Codes |
| title_full_unstemmed | Matrix-Product Codes over Commutative Rings and Constructions Arising from σ,δ-Codes |
| title_short | Matrix-Product Codes over Commutative Rings and Constructions Arising from σ,δ-Codes |
| title_sort | matrix product codes over commutative rings and constructions arising from σ δ codes |
| url | http://dx.doi.org/10.1155/2021/5521067 |
| work_keys_str_mv | AT mhammedboulagouaz matrixproductcodesovercommutativeringsandconstructionsarisingfromsdcodes AT abdulazizdeajim matrixproductcodesovercommutativeringsandconstructionsarisingfromsdcodes |