Optimal \((2,\delta)\) Locally Repairable Codes via Punctured Simplex Codes
Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal \((2, \delta)\)-LRCs over \(\mathbb{F}_q\) with flexible parameters. Firstly, employing techniques from finite geometry,...
Gespeichert in:
| Veröffentlicht in: | arXiv.org 2024-06 |
|---|---|
| Hauptverfasser: | , , , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | |
|---|---|
| container_issue | |
| container_start_page | |
| container_title | arXiv.org |
| container_volume | |
| creator | Gao, Yuan Fang, Weijun Xu, Jingke Wang, Dong Hu, Sihuang |
| description | Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal \((2, \delta)\)-LRCs over \(\mathbb{F}_q\) with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a \((2, \delta)\)-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes. |
| doi_str_mv | 10.48550/arxiv.2307.04323 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_arxiv</sourceid><recordid>TN_cdi_arxiv_primary_2307_04323</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2835677634</sourcerecordid><originalsourceid>FETCH-LOGICAL-a953-d5b9f48e4602fd1485dd7cb2d506867096e9b31750ded006b1984c40c5c44a963</originalsourceid><addsrcrecordid>eNotj81Kw0AYRQdBsNQ-gCsH3LRg4pf5zSwlWBUDFe0yECaZCaRMmzhJSvv2xsbVXdzL5RyE7iIIWcw5PGl_qo8hoSBDYJTQKzQjlEZBzAi5QYuu2wEAEZJwTmfoY9P29V47nC2X5DEz1vV6la1w2pTauTP-sq2uvS6cxUljbIePtcafw6HsB28N_q73rbOnqbtF15V2nV385xxt1y_b5C1IN6_vyXMaaMVpYHihKhZbJoBUJhqhjZFlQQwHEQsJSlhV0EhyMNYAiCJSMSsZlLxkTCtB5-h-ur2Y5q0f-f05_zPOL8bj4mFatL75GWzX57tm8IeRKScx5UJKQRn9BbRQVqI</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2835677634</pqid></control><display><type>article</type><title>Optimal \((2,\delta)\) Locally Repairable Codes via Punctured Simplex Codes</title><source>arXiv.org</source><source>Free E- Journals</source><creator>Gao, Yuan ; Fang, Weijun ; Xu, Jingke ; Wang, Dong ; Hu, Sihuang</creator><creatorcontrib>Gao, Yuan ; Fang, Weijun ; Xu, Jingke ; Wang, Dong ; Hu, Sihuang</creatorcontrib><description>Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal \((2, \delta)\)-LRCs over \(\mathbb{F}_q\) with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a \((2, \delta)\)-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes.</description><identifier>EISSN: 2331-8422</identifier><identifier>DOI: 10.48550/arxiv.2307.04323</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Computer Science - Information Theory ; Fields (mathematics) ; Mathematics - Information Theory ; Polynomials ; Storage systems</subject><ispartof>arXiv.org, 2024-06</ispartof><rights>2024. This work is published under http://creativecommons.org/publicdomain/zero/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><rights>http://creativecommons.org/publicdomain/zero/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,780,881,27902</link.rule.ids><backlink>$$Uhttps://doi.org/10.48550/arXiv.2307.04323$$DView paper in arXiv$$Hfree_for_read</backlink><backlink>$$Uhttps://doi.org/10.1007/s10623-024-01470-2$$DView published paper (Access to full text may be restricted)$$Hfree_for_read</backlink></links><search><creatorcontrib>Gao, Yuan</creatorcontrib><creatorcontrib>Fang, Weijun</creatorcontrib><creatorcontrib>Xu, Jingke</creatorcontrib><creatorcontrib>Wang, Dong</creatorcontrib><creatorcontrib>Hu, Sihuang</creatorcontrib><title>Optimal \((2,\delta)\) Locally Repairable Codes via Punctured Simplex Codes</title><title>arXiv.org</title><description>Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal \((2, \delta)\)-LRCs over \(\mathbb{F}_q\) with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a \((2, \delta)\)-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes.</description><subject>Computer Science - Information Theory</subject><subject>Fields (mathematics)</subject><subject>Mathematics - Information Theory</subject><subject>Polynomials</subject><subject>Storage systems</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>BENPR</sourceid><sourceid>GOX</sourceid><recordid>eNotj81Kw0AYRQdBsNQ-gCsH3LRg4pf5zSwlWBUDFe0yECaZCaRMmzhJSvv2xsbVXdzL5RyE7iIIWcw5PGl_qo8hoSBDYJTQKzQjlEZBzAi5QYuu2wEAEZJwTmfoY9P29V47nC2X5DEz1vV6la1w2pTauTP-sq2uvS6cxUljbIePtcafw6HsB28N_q73rbOnqbtF15V2nV385xxt1y_b5C1IN6_vyXMaaMVpYHihKhZbJoBUJhqhjZFlQQwHEQsJSlhV0EhyMNYAiCJSMSsZlLxkTCtB5-h-ur2Y5q0f-f05_zPOL8bj4mFatL75GWzX57tm8IeRKScx5UJKQRn9BbRQVqI</recordid><startdate>20240619</startdate><enddate>20240619</enddate><creator>Gao, Yuan</creator><creator>Fang, Weijun</creator><creator>Xu, Jingke</creator><creator>Wang, Dong</creator><creator>Hu, Sihuang</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>AKY</scope><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20240619</creationdate><title>Optimal \((2,\delta)\) Locally Repairable Codes via Punctured Simplex Codes</title><author>Gao, Yuan ; Fang, Weijun ; Xu, Jingke ; Wang, Dong ; Hu, Sihuang</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a953-d5b9f48e4602fd1485dd7cb2d506867096e9b31750ded006b1984c40c5c44a963</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Computer Science - Information Theory</topic><topic>Fields (mathematics)</topic><topic>Mathematics - Information Theory</topic><topic>Polynomials</topic><topic>Storage systems</topic><toplevel>online_resources</toplevel><creatorcontrib>Gao, Yuan</creatorcontrib><creatorcontrib>Fang, Weijun</creatorcontrib><creatorcontrib>Xu, Jingke</creatorcontrib><creatorcontrib>Wang, Dong</creatorcontrib><creatorcontrib>Hu, Sihuang</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>arXiv Computer Science</collection><collection>arXiv Mathematics</collection><collection>arXiv.org</collection><jtitle>arXiv.org</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Gao, Yuan</au><au>Fang, Weijun</au><au>Xu, Jingke</au><au>Wang, Dong</au><au>Hu, Sihuang</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Optimal \((2,\delta)\) Locally Repairable Codes via Punctured Simplex Codes</atitle><jtitle>arXiv.org</jtitle><date>2024-06-19</date><risdate>2024</risdate><eissn>2331-8422</eissn><abstract>Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal \((2, \delta)\)-LRCs over \(\mathbb{F}_q\) with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a \((2, \delta)\)-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><doi>10.48550/arxiv.2307.04323</doi><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | EISSN: 2331-8422 |
| ispartof | arXiv.org, 2024-06 |
| issn | 2331-8422 |
| language | eng |
| recordid | cdi_arxiv_primary_2307_04323 |
| source | arXiv.org; Free E- Journals |
| subjects | Computer Science - Information Theory Fields (mathematics) Mathematics - Information Theory Polynomials Storage systems |
| title | Optimal \((2,\delta)\) Locally Repairable Codes via Punctured Simplex Codes |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-02-02T16%3A14%3A44IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_arxiv&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Optimal%20%5C((2,%5Cdelta)%5C)%20Locally%20Repairable%20Codes%20via%20Punctured%20Simplex%20Codes&rft.jtitle=arXiv.org&rft.au=Gao,%20Yuan&rft.date=2024-06-19&rft.eissn=2331-8422&rft_id=info:doi/10.48550/arxiv.2307.04323&rft_dat=%3Cproquest_arxiv%3E2835677634%3C/proquest_arxiv%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2835677634&rft_id=info:pmid/&rfr_iscdi=true |