{"id":"https://openalex.org/W1975104559","doi":"https://doi.org/10.1109/netcod.2012.6261906","title":"The precoding problem for sparse random linear network coding","display_name":"The precoding problem for sparse random linear network coding","publication_year":2012,"publication_date":"2012-06-01","ids":{"openalex":"https://openalex.org/W1975104559","doi":"https://doi.org/10.1109/netcod.2012.6261906","mag":"1975104559"},"language":"en","primary_location":{"id":"doi:10.1109/netcod.2012.6261906","is_oa":false,"landing_page_url":"https://doi.org/10.1109/netcod.2012.6261906","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2012 International Symposium on Network Coding (NetCod)","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5107305990","display_name":"Xiaolin Li","orcid":"https://orcid.org/0000-0003-1360-2193"},"institutions":[{"id":"https://openalex.org/I200769079","display_name":"Hong Kong University of Science and Technology","ror":"https://ror.org/00q4vv597","country_code":"HK","type":"education","lineage":["https://openalex.org/I200769079"]}],"countries":["HK"],"is_corresponding":true,"raw_author_name":"Xiaolin Li","raw_affiliation_strings":["Hong Kong University of Science and Technology, Hong Kong, China","Hong Kong University of Science & Technology"],"affiliations":[{"raw_affiliation_string":"Hong Kong University of Science and Technology, Hong Kong, China","institution_ids":["https://openalex.org/I200769079"]},{"raw_affiliation_string":"Hong Kong University of Science & Technology","institution_ids":["https://openalex.org/I200769079"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5065074102","display_name":"Wai Ho Mow","orcid":"https://orcid.org/0000-0003-1804-0476"},"institutions":[{"id":"https://openalex.org/I200769079","display_name":"Hong Kong University of Science and Technology","ror":"https://ror.org/00q4vv597","country_code":"HK","type":"education","lineage":["https://openalex.org/I200769079"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Wai Ho Mow","raw_affiliation_strings":["Hong Kong University of Science and Technology, Hong Kong, China","Hong Kong University of Science & Technology"],"affiliations":[{"raw_affiliation_string":"Hong Kong University of Science and Technology, Hong Kong, China","institution_ids":["https://openalex.org/I200769079"]},{"raw_affiliation_string":"Hong Kong University of Science & Technology","institution_ids":["https://openalex.org/I200769079"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5107305990"],"corresponding_institution_ids":["https://openalex.org/I200769079"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.09146919,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"185","last_page":"190"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10796","display_name":"Cooperative Communication and Network Coding","score":1.0,"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":1.0,"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":0.9731000065803528,"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":0.9627000093460083,"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/linear-network-coding","display_name":"Linear network coding","score":0.7556437253952026},{"id":"https://openalex.org/keywords/precoding","display_name":"Precoding","score":0.6570056080818176},{"id":"https://openalex.org/keywords/decoding-methods","display_name":"Decoding methods","score":0.5783472657203674},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.5714277029037476},{"id":"https://openalex.org/keywords/generator-matrix","display_name":"Generator matrix","score":0.5390353202819824},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.49443507194519043},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4684239625930786},{"id":"https://openalex.org/keywords/multivariate-random-variable","display_name":"Multivariate random variable","score":0.46306318044662476},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4427134394645691},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4189397096633911},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.41655534505844116},{"id":"https://openalex.org/keywords/linear-code","display_name":"Linear code","score":0.4159245789051056},{"id":"https://openalex.org/keywords/network-packet","display_name":"Network packet","score":0.4089231789112091},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.25682052969932556},{"id":"https://openalex.org/keywords/block-code","display_name":"Block code","score":0.25563880801200867},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.22381746768951416},{"id":"https://openalex.org/keywords/mimo","display_name":"MIMO","score":0.18476837873458862},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.09431120753288269},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.09217575192451477}],"concepts":[{"id":"https://openalex.org/C138293262","wikidata":"https://www.wikidata.org/wiki/Q1089578","display_name":"Linear network coding","level":3,"score":0.7556437253952026},{"id":"https://openalex.org/C160562895","wikidata":"https://www.wikidata.org/wiki/Q7239557","display_name":"Precoding","level":4,"score":0.6570056080818176},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.5783472657203674},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.5714277029037476},{"id":"https://openalex.org/C1842014","wikidata":"https://www.wikidata.org/wiki/Q1502046","display_name":"Generator matrix","level":3,"score":0.5390353202819824},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.49443507194519043},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4684239625930786},{"id":"https://openalex.org/C138405894","wikidata":"https://www.wikidata.org/wiki/Q3179949","display_name":"Multivariate random variable","level":3,"score":0.46306318044662476},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4427134394645691},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4189397096633911},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.41655534505844116},{"id":"https://openalex.org/C2400350","wikidata":"https://www.wikidata.org/wiki/Q1752667","display_name":"Linear code","level":4,"score":0.4159245789051056},{"id":"https://openalex.org/C158379750","wikidata":"https://www.wikidata.org/wiki/Q214111","display_name":"Network packet","level":2,"score":0.4089231789112091},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.25682052969932556},{"id":"https://openalex.org/C157125643","wikidata":"https://www.wikidata.org/wiki/Q884707","display_name":"Block code","level":3,"score":0.25563880801200867},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.22381746768951416},{"id":"https://openalex.org/C207987634","wikidata":"https://www.wikidata.org/wiki/Q176862","display_name":"MIMO","level":3,"score":0.18476837873458862},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.09431120753288269},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.09217575192451477},{"id":"https://openalex.org/C54197355","wikidata":"https://www.wikidata.org/wiki/Q5782992","display_name":"Beamforming","level":2,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1109/netcod.2012.6261906","is_oa":false,"landing_page_url":"https://doi.org/10.1109/netcod.2012.6261906","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2012 International Symposium on Network Coding (NetCod)","raw_type":"proceedings-article"},{"id":"pmh:oai:repository.hkust.edu.hk:1783.1-52563","is_oa":false,"landing_page_url":"http://repository.hkust.edu.hk/ir/Record/1783.1-52563","pdf_url":null,"source":{"id":"https://openalex.org/S4306401796","display_name":"Rare & Special e-Zone (The Hong Kong University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I200769079","host_organization_name":"Hong Kong University of Science and Technology","host_organization_lineage":["https://openalex.org/I200769079"],"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":"Conference paper"},{"id":"pmh:oai:repository.ust.hk:1783.1-52563","is_oa":false,"landing_page_url":"http://www.scopus.com/record/display.url?eid=2-s2.0-84866690812&origin=inward","pdf_url":null,"source":{"id":"https://openalex.org/S4306401796","display_name":"Rare & Special e-Zone (The Hong Kong University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I200769079","host_organization_name":"Hong Kong University of Science and Technology","host_organization_lineage":["https://openalex.org/I200769079"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Conference paper"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":15,"referenced_works":["https://openalex.org/W1606480398","https://openalex.org/W1914271854","https://openalex.org/W1966612101","https://openalex.org/W1971862312","https://openalex.org/W2048235391","https://openalex.org/W2095022281","https://openalex.org/W2104812861","https://openalex.org/W2105831729","https://openalex.org/W2122912623","https://openalex.org/W2137266817","https://openalex.org/W2139416652","https://openalex.org/W2140865874","https://openalex.org/W2155398259","https://openalex.org/W2165084253","https://openalex.org/W6640182316"],"related_works":["https://openalex.org/W4247927056","https://openalex.org/W2131338819","https://openalex.org/W4288278449","https://openalex.org/W2966039054","https://openalex.org/W1487470402","https://openalex.org/W4237142313","https://openalex.org/W2099549796","https://openalex.org/W2974676816","https://openalex.org/W2052380558","https://openalex.org/W2373785022"],"abstract_inverted_index":{"In":[0],"the":[1,12,18,23,26,28,37,47,53,71,78,89,111,120,136,139,144,148,151],"random":[2,96,117],"linear":[3],"network":[4,19],"coding":[5],"scenario":[6],"with":[7],"subspace":[8,152],"codes":[9,82],"applied,":[10],"if":[11],"number":[13],"of":[14,25,39,74,77,114,143,150],"packets":[15,29,44],"injected":[16],"into":[17],"is":[20,107,129],"larger":[21],"than":[22],"dimension":[24],"subspace,":[27],"are":[30],"not":[31],"linearly":[32],"independent.":[33],"This":[34],"paper":[35],"addresses":[36],"problem":[38],"how":[40],"to":[41,45,51,131],"choose":[42],"these":[43],"represent":[46],"subspaces":[48],"so":[49],"as":[50,60],"minimize":[52,88],"decoding":[54,90],"failure":[55,91],"probability":[56,92],"and":[57,83,147],"formulates":[58],"it":[59,86,128],"a":[61,66,75,94,100,126],"precoding":[62,67],"problem.":[63],"We":[64],"propose":[65],"method":[68],"based":[69],"on":[70,135],"generator":[72],"matrices":[73,118],"class":[76],"maximum":[79,140],"distance":[80],"separable":[81],"show":[84],"that":[85],"can":[87],"for":[93],"sparse":[95],"transfer":[97,145],"matrix":[98,146],"over":[99],"large":[101,121],"enough":[102],"finite":[103,115],"field.":[104],"Our":[105],"result":[106],"obtained":[108],"by":[109],"analyzing":[110],"rank":[112],"distribution":[113],"field":[116,122],"in":[119],"size":[123],"limit.":[124],"As":[125],"consequence,":[127],"applied":[130],"shed":[132],"some":[133],"light":[134],"tradeoff":[137],"between":[138],"achievable":[141],"sparsity":[142],"rate":[149],"code.":[153]},"counts_by_year":[],"updated_date":"2026-02-25T08:12:03.925757","created_date":"2025-10-10T00:00:00"}