{"id":"https://openalex.org/W2477854368","doi":"https://doi.org/10.1109/tit.2017.2697421","title":"Linear Network Coding Over Rings \u2013 Part I: Scalar Codes and Commutative Alphabets","display_name":"Linear Network Coding Over Rings \u2013 Part I: Scalar Codes and Commutative Alphabets","publication_year":2017,"publication_date":"2017-04-24","ids":{"openalex":"https://openalex.org/W2477854368","doi":"https://doi.org/10.1109/tit.2017.2697421","mag":"2477854368"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2017.2697421","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2017.2697421","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1608.01738","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5082875389","display_name":"Joseph Connelly","orcid":"https://orcid.org/0000-0001-7307-7023"},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California, San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Joseph Connelly","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of California at San Diego, San Diego, La Jolla, CA, USA"],"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of California at San Diego, San Diego, La Jolla, CA, USA","institution_ids":["https://openalex.org/I36258959"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072111370","display_name":"K. Zeger","orcid":"https://orcid.org/0000-0001-6415-1447"},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California, San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kenneth Zeger","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of California at San Diego, San Diego, La Jolla, CA, USA"],"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of California at San Diego, San Diego, La Jolla, CA, USA","institution_ids":["https://openalex.org/I36258959"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5082875389"],"corresponding_institution_ids":["https://openalex.org/I36258959"],"apc_list":null,"apc_paid":null,"fwci":2.1895,"has_fulltext":false,"cited_by_count":17,"citation_normalized_percentile":{"value":0.88927172,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"64","issue":"1","first_page":"274","last_page":"291"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10796","display_name":"Cooperative Communication and Network Coding","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10796","display_name":"Cooperative Communication and Network Coding","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12791","display_name":"Full-Duplex Wireless Communications","score":0.9506000280380249,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11458","display_name":"Advanced Wireless Communication Technologies","score":0.9366999864578247,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7578806281089783},{"id":"https://openalex.org/keywords/commutative-ring","display_name":"Commutative ring","score":0.6820695400238037},{"id":"https://openalex.org/keywords/commutative-property","display_name":"Commutative property","score":0.6362482309341431},{"id":"https://openalex.org/keywords/linear-network-coding","display_name":"Linear network coding","score":0.5375518202781677},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5259318947792053},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.4754611849784851},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.42115098237991333},{"id":"https://openalex.org/keywords/polynomial-ring","display_name":"Polynomial ring","score":0.4192306399345398},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3544699549674988},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12477952241897583},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.08066588640213013},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.05820402503013611}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7578806281089783},{"id":"https://openalex.org/C161491579","wikidata":"https://www.wikidata.org/wiki/Q858656","display_name":"Commutative ring","level":3,"score":0.6820695400238037},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.6362482309341431},{"id":"https://openalex.org/C138293262","wikidata":"https://www.wikidata.org/wiki/Q1089578","display_name":"Linear network coding","level":3,"score":0.5375518202781677},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5259318947792053},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.4754611849784851},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.42115098237991333},{"id":"https://openalex.org/C9485509","wikidata":"https://www.wikidata.org/wiki/Q1455652","display_name":"Polynomial ring","level":3,"score":0.4192306399345398},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3544699549674988},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12477952241897583},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.08066588640213013},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.05820402503013611},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0},{"id":"https://openalex.org/C158379750","wikidata":"https://www.wikidata.org/wiki/Q214111","display_name":"Network packet","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tit.2017.2697421","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2017.2697421","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1608.01738","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1608.01738","pdf_url":"https://arxiv.org/pdf/1608.01738","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1608.01738","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1608.01738","pdf_url":"https://arxiv.org/pdf/1608.01738","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":45,"referenced_works":["https://openalex.org/W110310473","https://openalex.org/W1483151078","https://openalex.org/W1553920270","https://openalex.org/W1730126071","https://openalex.org/W1804057676","https://openalex.org/W1939076699","https://openalex.org/W2001466972","https://openalex.org/W2016444590","https://openalex.org/W2021430885","https://openalex.org/W2028078858","https://openalex.org/W2038553845","https://openalex.org/W2048235391","https://openalex.org/W2049700673","https://openalex.org/W2052813097","https://openalex.org/W2068263566","https://openalex.org/W2090966880","https://openalex.org/W2102203742","https://openalex.org/W2103320637","https://openalex.org/W2104750217","https://openalex.org/W2105831729","https://openalex.org/W2106403318","https://openalex.org/W2117872622","https://openalex.org/W2123562143","https://openalex.org/W2130512442","https://openalex.org/W2132123317","https://openalex.org/W2138928022","https://openalex.org/W2167703619","https://openalex.org/W2168674495","https://openalex.org/W2176518791","https://openalex.org/W2262362834","https://openalex.org/W2295046799","https://openalex.org/W2315729195","https://openalex.org/W2475505336","https://openalex.org/W2525693147","https://openalex.org/W2532691608","https://openalex.org/W2561436563","https://openalex.org/W2791874884","https://openalex.org/W2963428994","https://openalex.org/W2963599681","https://openalex.org/W3098342432","https://openalex.org/W3099798328","https://openalex.org/W3100411620","https://openalex.org/W3101064986","https://openalex.org/W4231614163","https://openalex.org/W6654396071"],"related_works":["https://openalex.org/W2962934782","https://openalex.org/W2964326893","https://openalex.org/W1978343121","https://openalex.org/W4308298797","https://openalex.org/W4255963200","https://openalex.org/W4312973762","https://openalex.org/W2360590989","https://openalex.org/W210140635","https://openalex.org/W2021515234","https://openalex.org/W2035028562"],"abstract_inverted_index":{"Linear":[0],"network":[1,35,68,157,252],"coding":[2,18,36],"over":[3,37,74,106,115,162,167,257,266],"finite":[4,24,76,238],"fields":[5,40,188],"is":[6,49,61,86,150,158,208,240,253],"a":[7,67,70,87,141,147,209,242,258],"wellstudied":[8],"problem.":[9],"We":[10,63,89,119,144],"consider":[11],"the":[12,46,58,80,101,136,163,173,191,195,200,267,270],"more":[13],"general":[14],"setting":[15],"of":[16,140,172,186,194,199,214,224,261,269],"linear":[17,34,72],"for":[19,32,247],"directed":[20],"acyclic":[21],"networks":[22,105],"with":[23,126],"commutative":[25,38,77,84,94,121,138,148,170,181,212,259],"ring":[26,85,109,149,171,213,244,260],"alphabets.":[27],"Our":[28],"results":[29],"imply":[30],"that":[31,65,92,99,123,146,179,206],"scalar":[33,71,102,112,159,254],"rings,":[39],"can":[41],"always":[42],"be":[43,55,134],"used":[44],"when":[45,57],"alphabet":[47,59],"size":[48,60,215,262,272],"flexible,":[50],"but":[51,165,264],"other":[52,169],"rings":[53,95,122,139,182],"may":[54,133],"needed":[56],"fixed.":[62],"prove":[64,145,205],"if":[66,152,155,217,220,273,276],"has":[69,226],"solution":[73],"some":[75,156,251],"ring,":[78,164],"then":[79],"(unique)":[81],"smallest":[82],"such":[83,98,241],"field.":[88],"also":[90,111],"show":[91,178],"fixed-size":[93],"are":[96,110,124,183],"quasi-ordered,":[97],"all":[100],"linearly":[103,113,160,255],"solvable":[104,114,161,256],"any":[107,116,168],"given":[108,142],"higher-ordered":[117],"ring.":[118],"study":[120],"maximal":[125,151,180,211,243],"respect":[127],"to":[128],"this":[129],"quasiorder,":[130],"as":[131],"they":[132],"considered":[135],"best":[137],"size.":[143,175,202],"and":[153,218,245,274],"only":[154,219,275],"not":[166,265],"same":[174,271],"Furthermore,":[176],"we":[177,204],"direct":[184],"products":[185],"certain":[187],"specified":[189],"by":[190],"integer":[192],"partitions":[193],"prime":[196,222,249],"factor":[197,223],"multiplicities":[198],"ring's":[201],"Finally,":[203],"there":[207],"unique":[210],"m":[216,225],"each":[221,248],"multiplicity":[227],"in":[228],"{1,":[229,279],"2,":[230,280],"3,":[231,281],"4,":[232,282],"6}.":[233,283],"As":[234],"consequences,":[235],"1)":[236],"every":[237],"field":[239,268],"2)":[246],"p,":[250],"pk":[263],"k":[277],"\u2209":[278]},"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2016-08-23T00:00:00"}