{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:06:07Z","timestamp":1760234767629,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,6,17]],"date-time":"2021-06-17T00:00:00Z","timestamp":1623888000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministerio de Ciencia e Innovaci\u00f3n of the Gobierno de Espa\u00f1a","award":["PID2019-108668GB-I00"],"award-info":[{"award-number":["PID2019-108668GB-I00"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>In this paper, we study product convolutional codes described by state-space representations. In particular, we investigate how to derive state-space representations of the product code from the horizontal and vertical convolutional codes. We present a systematic procedure to build such representation with minimal dimension, i.e., reachable and observable.<\/jats:p>","DOI":"10.3390\/math9121410","type":"journal-article","created":{"date-parts":[[2021,6,17]],"date-time":"2021-06-17T11:20:26Z","timestamp":1623928826000},"page":"1410","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Minimal State-Space Representation of Convolutional Product Codes"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0522-0304","authenticated-orcid":false,"given":"Joan-Josep","family":"Climent","sequence":"first","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat d\u2019Alacant, E-03690 Alacant, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2303-9824","authenticated-orcid":false,"given":"Diego","family":"Napp","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat d\u2019Alacant, E-03690 Alacant, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8168-4023","authenticated-orcid":false,"given":"Raquel","family":"Pinto","sequence":"additional","affiliation":[{"name":"CIDMA\u2014Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1497-6456","authenticated-orcid":false,"given":"Ver\u00f3nica","family":"Requena","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat d\u2019Alacant, E-03690 Alacant, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1109\/12.364531","article-title":"EVENODD: An efficient scheme for tolerating double disk failures in RAID architectures","volume":"42","author":"Blaum","year":"1995","journal-title":"IEEE Trans. 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