{"id":"https://openalex.org/W7213288595","doi":"https://doi.org/10.48550/arxiv.2609.14434","title":"Hermitian Duality and Syndrome Structure of CSS-Type Codes over the ring $\\mathbb{Z}_q[i]$","display_name":"Hermitian Duality and Syndrome Structure of CSS-Type Codes over the ring $\\mathbb{Z}_q[i]$","publication_year":2026,"publication_date":"2026-09-13","ids":{"openalex":"https://openalex.org/W7213288595","doi":"https://doi.org/10.48550/arxiv.2609.14434"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.14434","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14434","pdf_url":"https://arxiv.org/pdf/2609.14434","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.14434","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5118970194","display_name":"Akanksha Tiwari","orcid":"https://orcid.org/0009-0006-4407-5700","observed_orcids":["https://orcid.org/0009-0006-4407-5700"]},"institutions":[{"id":"https://openalex.org/I68891433","display_name":"Indian Institute of Technology Delhi","ror":"https://ror.org/049tgcd06","country_code":"IN","type":"education","lineage":["https://openalex.org/I68891433"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Tiwari, Akanksha","raw_affiliation_strings":["Department of Mathematics Indian Institute of Technology Delhi Hauz Khas , New Delhi- 110016 , India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics Indian Institute of Technology Delhi Hauz Khas , New Delhi- 110016 , India","institution_ids":["https://openalex.org/I68891433"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5035273961","display_name":"Ritumoni Sarma","orcid":"https://orcid.org/0000-0002-8668-1948","observed_orcids":["https://orcid.org/0000-0002-8668-1948"]},"institutions":[{"id":"https://openalex.org/I68891433","display_name":"Indian Institute of Technology Delhi","ror":"https://ror.org/049tgcd06","country_code":"IN","type":"education","lineage":["https://openalex.org/I68891433"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Sarma, Ritumoni","raw_affiliation_strings":["Department of Mathematics Indian Institute of Technology Delhi Hauz Khas , New Delhi- 110016 , India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics Indian Institute of Technology Delhi Hauz Khas , New Delhi- 110016 , India","institution_ids":["https://openalex.org/I68891433"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I68891433"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.7935000061988831,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.7935000061988831,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.1981000006198883,"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.0015999999595806003,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"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/quantum-error-correction","display_name":"quantum error correction","score":0.7749999761581421},{"id":"https://openalex.org/keywords/nested-codes","display_name":"nested codes","score":0.7229999899864197},{"id":"https://openalex.org/keywords/annihilators","display_name":"annihilators","score":0.6890000104904175},{"id":"https://openalex.org/keywords/logical-connectives","display_name":"logical connectives","score":0.5580000281333923},{"id":"https://openalex.org/keywords/transversals","display_name":"transversals","score":0.5389999747276306},{"id":"https://openalex.org/keywords/syndrome-measurement","display_name":"syndrome measurement","score":0.4650000035762787}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7390999794006348},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.7315999865531921},{"id":"https://openalex.org/C78766492","wikidata":"https://www.wikidata.org/wiki/Q44382","display_name":"Principal ideal","level":3,"score":0.6309999823570251},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.5490999817848206},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4828999936580658},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.4634999930858612},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.4442000091075897},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.42980000376701355},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.4237000048160553},{"id":"https://openalex.org/C144559511","wikidata":"https://www.wikidata.org/wiki/Q2986279","display_name":"Principal (computer security)","level":2,"score":0.40450000762939453},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.38769999146461487},{"id":"https://openalex.org/C2777502316","wikidata":"https://www.wikidata.org/wiki/Q2712515","display_name":"Stabilizer (aeronautics)","level":2,"score":0.38350000977516174},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.3736000061035156},{"id":"https://openalex.org/C8367249","wikidata":"https://www.wikidata.org/wiki/Q912083","display_name":"Ideal class group","level":3,"score":0.3718000054359436},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.367000013589859},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3506999909877777},{"id":"https://openalex.org/C182419690","wikidata":"https://www.wikidata.org/wiki/Q205464","display_name":"Algebraic structure","level":2,"score":0.34700000286102295},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.33329999446868896},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.3255000114440918},{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.3160000145435333},{"id":"https://openalex.org/C3427991","wikidata":"https://www.wikidata.org/wiki/Q651426","display_name":"Group code","level":5,"score":0.29330000281333923},{"id":"https://openalex.org/C2780130068","wikidata":"https://www.wikidata.org/wiki/Q2208651","display_name":"Transversal (combinatorics)","level":2,"score":0.29159998893737793},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.2842999994754791},{"id":"https://openalex.org/C2779467367","wikidata":"https://www.wikidata.org/wiki/Q863912","display_name":"Prime ideal","level":3,"score":0.2793000042438507},{"id":"https://openalex.org/C2776196297","wikidata":"https://www.wikidata.org/wiki/Q17138781","display_name":"Twist","level":2,"score":0.27320000529289246},{"id":"https://openalex.org/C9485509","wikidata":"https://www.wikidata.org/wiki/Q1455652","display_name":"Polynomial ring","level":3,"score":0.26579999923706055},{"id":"https://openalex.org/C2780791683","wikidata":"https://www.wikidata.org/wiki/Q846785","display_name":"Action (physics)","level":2,"score":0.2635999917984009},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.262800008058548},{"id":"https://openalex.org/C2989065042","wikidata":"https://www.wikidata.org/wiki/Q7378971","display_name":"Algebraic properties","level":2,"score":0.2549000084400177},{"id":"https://openalex.org/C156923205","wikidata":"https://www.wikidata.org/wiki/Q2905043","display_name":"Principal ideal ring","level":4,"score":0.2531000077724457}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.14434","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14434","pdf_url":"https://arxiv.org/pdf/2609.14434","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},{"id":"doi:10.48550/arxiv.2609.14434","doi":"https://doi.org/10.48550/arxiv.2609.14434","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.14434","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.14434","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14434","pdf_url":"https://arxiv.org/pdf/2609.14434","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213288595.pdf","grobid_xml":"https://content.openalex.org/works/W7213288595.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7170087662","https://openalex.org/W7162670066","https://openalex.org/W4402615390","https://openalex.org/W2259108228","https://openalex.org/W4404698122","https://openalex.org/W7168393492","https://openalex.org/W2528899522","https://openalex.org/W7124433795","https://openalex.org/W7167357901","https://openalex.org/W4399330707"],"abstract_inverted_index":{"We":[0,66],"study":[1],"the":[2,14,40,48,104,111,114,176,182],"algebraic":[3,57],"structure":[4],"of":[5,23,42,113,144],"principal":[6,52,71,79],"ideal":[7,53,80],"codes":[8,95],"and":[9,44,59,73,108,129,186],"CSS-type":[10,106],"stabilizer":[11,60,107,164,177],"constructions":[12,34],"over":[13],"ring":[15],"$":[16,90,92,97,101,120,122],"R_q=\\mathbb":[17],"Z_q[i],$":[18],"where":[19],"every":[20],"prime":[21],"divisor":[22],"$q$":[24],"is":[25,89,119,161,173],"congruent":[26],"to":[27],"$1$":[28],"modulo":[29],"$4$.":[30],"Although":[31],"such":[32],"ring-based":[33],"have":[35,150],"recently":[36],"been":[37],"considered":[38],"in":[39,146],"context":[41],"classical":[43],"quantum":[45],"error":[46,159],"correction,":[47],"precise":[49],"relationship":[50],"among":[51],"cardinalities,":[54],"Hermitian":[55,87],"duality,":[56,184],"cosets,":[58],"syndromes":[61],"requires":[62],"further":[63],"structural":[64],"analysis.":[65],"obtain":[67],"cardinality":[68],"formulas":[69],"for":[70],"ideals":[72],"their":[74],"annihilators.":[75],"For":[76,93],"a":[77,142,168],"length-one":[78],"code":[81],"$C=\\langle\u03b1\\rangle$,":[82],"we":[83,102,140],"show":[84],"that":[85,110,172],"its":[86],"dual":[88],"C^{\\perp_H}=\\operatorname{Ann}(\u03c3(\u03b1)).":[91],"nested":[94],"satisfying":[96],"C_2^{\\perp_H}\\subseteq":[98],"C_1\\subseteq":[99],"C_2,":[100],"determine":[103],"corresponding":[105,157],"prove":[109],"kernel":[112],"physical":[115,132,153],"$X$-error":[116,133,154],"syndrome":[117,134,138,165,187],"map":[118],"\\ker(\\operatorname{Syn}_X)=C_2.":[121],"Consequently,":[123],"$C_2/C_1$":[124],"parametrizes":[125,131],"logical":[126],"$X$-operator":[127],"classes":[128],"$R_q^n/C_2$":[130],"classes.":[135],"Using":[136],"this":[137],"quotient,":[139],"construct":[141],"transversal":[143],"$C_2$":[145],"$R_q^n$":[147],"whose":[148],"elements":[149],"pairwise":[151],"distinct":[152],"syndromes.":[155],"The":[156],"$X$-type":[158],"family":[160],"correctable":[162],"by":[163],"measurement,":[166],"yielding":[167],"syndrome-based":[169],"recovery":[170],"procedure":[171],"consistent":[174],"with":[175],"structure.":[178,188],"Explicit":[179],"examples":[180],"illustrate":[181],"resulting":[183],"cardinality,":[185]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-16T00:00:00"}