{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:04:28Z","timestamp":1760241868627,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2018,10,24]],"date-time":"2018-10-24T00:00:00Z","timestamp":1540339200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 61371078"],"award-info":[{"award-number":["No. 61371078"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"<jats:p>Locally repairable codes (LRCs) are a new family of erasure codes used in distributed storage systems which have attracted a great deal of interest in recent years. For an     [ n , k , d ]     linear code, if a code symbol can be repaired by t disjoint groups of other code symbols, where each group contains at most r code symbols, it is said to have availability-    ( r , t )    . Single-parity LRCs are LRCs with a constraint that each repairable group contains exactly one parity symbol. For an     [ n , k , d ]     single-parity LRC with availability-    ( r , t )     for the information symbols (single-parity LRCs), the minimum distance satisfies     d \u2264 n \u2212 k \u2212 \u2308 k t \/ r \u2309 + t + 1    . In this paper, we focus on the study of single-parity LRCs with availability-    ( r , t )     for information symbols. Based on the standard form of generator matrices, we present a novel characterization of single-parity LRCs with availability     t \u2265 1    . Then, a simple and straightforward proof for the Singleton-type bound is given based on the new characterization. Some necessary conditions for optimal single-parity LRCs with availability     t \u2265 1     are obtained, which might provide some guidelines for optimal coding constructions.<\/jats:p>","DOI":"10.3390\/info9110265","type":"journal-article","created":{"date-parts":[[2018,10,24]],"date-time":"2018-10-24T10:40:48Z","timestamp":1540377648000},"page":"265","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Single-Parity Locally Repairable Codes with Multiple Repairable Groups"],"prefix":"10.3390","volume":"9","author":[{"given":"Yanbo","family":"Lu","sequence":"first","affiliation":[{"name":"Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China"},{"name":"Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China"}]},{"given":"Xinji","family":"Liu","sequence":"additional","affiliation":[{"name":"Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China"},{"name":"Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China"}]},{"given":"Shutao","family":"Xia","sequence":"additional","affiliation":[{"name":"Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China"},{"name":"Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,24]]},"reference":[{"unstructured":"Huang, C., Simitci, H., Xu, Y., Ogus, A., Calder, B., Gopalan, P., Li, J., and Yekhanin, S. 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