{"id":"https://openalex.org/W3135006730","doi":"https://doi.org/10.3934/amc.2020135","title":"Duadic codes over &lt;inline-formula&gt;&lt;tex-math id=\"M1\"&gt;$ \\mathbb{Z}_4+u\\mathbb{Z}_4 $&lt;/tex-math&gt;&lt;/inline-formula&gt;","display_name":"Duadic codes over &lt;inline-formula&gt;&lt;tex-math id=\"M1\"&gt;$ \\mathbb{Z}_4+u\\mathbb{Z}_4 $&lt;/tex-math&gt;&lt;/inline-formula&gt;","publication_year":2021,"publication_date":"2021-03-08","ids":{"openalex":"https://openalex.org/W3135006730","doi":"https://doi.org/10.3934/amc.2020135","mag":"3135006730"},"language":"en","primary_location":{"id":"doi:10.3934/amc.2020135","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2020135","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=c3fa0a3a-0404-4dd6-b0cc-231fb3965dfa","source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.aimsciences.org/data/article/export-pdf?id=c3fa0a3a-0404-4dd6-b0cc-231fb3965dfa","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5050698107","display_name":"Raj Kumar","orcid":"https://orcid.org/0000-0002-4086-5194"},"institutions":[{"id":"https://openalex.org/I154851008","display_name":"Indian Institute of Technology Roorkee","ror":"https://ror.org/00582g326","country_code":"IN","type":"education","lineage":["https://openalex.org/I154851008"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Raj Kumar","raw_affiliation_strings":["Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India","institution_ids":["https://openalex.org/I154851008"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073177022","display_name":"Maheshanand Bhaintwal","orcid":"https://orcid.org/0000-0002-1471-5512"},"institutions":[{"id":"https://openalex.org/I154851008","display_name":"Indian Institute of Technology Roorkee","ror":"https://ror.org/00582g326","country_code":"IN","type":"education","lineage":["https://openalex.org/I154851008"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Maheshanand Bhaintwal","raw_affiliation_strings":["Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India","institution_ids":["https://openalex.org/I154851008"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5073177022"],"corresponding_institution_ids":["https://openalex.org/I154851008"],"apc_list":null,"apc_paid":null,"fwci":0.14,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.5238305,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"17","issue":"2","first_page":"404","last_page":"417"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11130","display_name":"Coding theory and cryptography","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11797","display_name":"graph theory and CDMA systems","score":0.9718999862670898,"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/T12162","display_name":"Cellular Automata and Applications","score":0.9559000134468079,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/abelian-group","display_name":"Abelian group","score":0.7191367149353027},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.691294252872467},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6604044437408447},{"id":"https://openalex.org/keywords/hamming-distance","display_name":"Hamming distance","score":0.4816228747367859},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40808600187301636}],"concepts":[{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.7191367149353027},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.691294252872467},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6604044437408447},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.4816228747367859},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40808600187301636}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3934/amc.2020135","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2020135","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=c3fa0a3a-0404-4dd6-b0cc-231fb3965dfa","source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3934/amc.2020135","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2020135","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=c3fa0a3a-0404-4dd6-b0cc-231fb3965dfa","source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G343156164","display_name":null,"funder_award_id":"India","funder_id":"https://openalex.org/F4320322724","funder_display_name":"Ministry of Education, India"}],"funders":[{"id":"https://openalex.org/F4320322724","display_name":"Ministry of Education, India","ror":"https://ror.org/048xjjh50"},{"id":"https://openalex.org/F4320337407","display_name":"Division of Human Resource Development","ror":"https://ror.org/03mamvh39"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3135006730.pdf","grobid_xml":"https://content.openalex.org/works/W3135006730.grobid-xml"},"referenced_works_count":20,"referenced_works":["https://openalex.org/W613035838","https://openalex.org/W1997847814","https://openalex.org/W2011039483","https://openalex.org/W2019389701","https://openalex.org/W2027330507","https://openalex.org/W2038681062","https://openalex.org/W2039466900","https://openalex.org/W2056472639","https://openalex.org/W2069531757","https://openalex.org/W2166231773","https://openalex.org/W2242376895","https://openalex.org/W2274326327","https://openalex.org/W2398151812","https://openalex.org/W2519285928","https://openalex.org/W2530905272","https://openalex.org/W2542760130","https://openalex.org/W2554589091","https://openalex.org/W2997721947","https://openalex.org/W3125735205","https://openalex.org/W6600659715"],"related_works":["https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W4225152035","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694","https://openalex.org/W2911598644","https://openalex.org/W2042127053"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"study":[4,62],"the":[5,56,89,113,122,142,190],"structure":[6],"of":[7,10,71,81,91,112,145,168,180],"duadic":[8,82,96,178],"codes":[9,64,83,97,111,167,179],"an":[11,146,154],"odd":[12],"length":[13,115,169,181],"$":[14,16,18,21,29,32,38,43,45,49,99,101,117,119,132,134,150,152,158,170,172,174,176,182,184,186,188,198,200,202],"n":[15],"over":[17,28,98,116,131,149,157,173,185],"\\mathbb{Z}_4+u\\mathbb{Z}_4":[19,100,118,133,151,187,199],"$,":[20,25,31,36,42],"u^2":[22,33],"=":[23,34,40,52],"0":[24,35],"(more":[26],"generally":[27],"\\mathbb{Z}_{q}+u\\mathbb{Z}_{q}":[30],"where":[37],"q":[39],"p^r":[41],"p":[44],"a":[46,69,106,194],"prime":[47],"and":[48,79,94,137,193],"(n,":[50],"p)":[51],"1":[53],"$)":[54],"using":[55],"discrete":[57],"Fourier":[58],"transform":[59],"approach.":[60],"We":[61,104,161],"these":[63],"by":[65],"considering":[66],"them":[67],"as":[68],"class":[70],"abelian":[72,110,147,155],"codes.":[73],"Some":[74,86],"results":[75],"related":[76],"to":[77,120,201],"self-duality":[78],"self-orthogonality":[80],"are":[84,102],"presented.":[85],"conditions":[87],"on":[88],"existence":[90],"self-dual":[92],"augmented":[93],"extended":[95],"determined.":[103],"present":[105],"sufficient":[107],"condition":[108],"for":[109],"same":[114,123],"have":[121,162],"minimum":[124],"Hamming":[125],"distance.":[126],"A":[127],"new":[128,165,195],"Gray":[129,143,191],"map":[130,192,196],"is":[135,139,153],"defined,":[136],"it":[138],"shown":[140],"that":[141],"image":[144],"code":[148,156],"\\mathbb{Z}_4":[159,175],"$.":[160,204],"obtained":[163],"five":[164],"linear":[166],"18":[171],"from":[177,197],"9":[183],"through":[189],"\\mathbb{Z}_4^2":[203]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
