{"id":"https://openalex.org/W7133220536","doi":"https://doi.org/10.48550/arxiv.2602.24243","title":"Finite Block Length Rate-Distortion Theory for the Bernoulli Source with Hamming Distortion: A Tutorial","display_name":"Finite Block Length Rate-Distortion Theory for the Bernoulli Source with Hamming Distortion: A Tutorial","publication_year":2026,"publication_date":"2026-02-27","ids":{"openalex":"https://openalex.org/W7133220536","doi":"https://doi.org/10.48550/arxiv.2602.24243"},"language":null,"primary_location":{"id":"pmh:doi:10.48550/arxiv.2602.24243","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5127824336","display_name":"Bhaskar Krishnamachari","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Krishnamachari, Bhaskar","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5127824336"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T10964","display_name":"Wireless Communication Security Techniques","score":0.33309999108314514,"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"}},"topics":[{"id":"https://openalex.org/T10964","display_name":"Wireless Communication Security Techniques","score":0.33309999108314514,"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/T10901","display_name":"Advanced Data Compression Techniques","score":0.2337999939918518,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.09700000286102295,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hamming-code","display_name":"Hamming code","score":0.6388000249862671},{"id":"https://openalex.org/keywords/lossy-compression","display_name":"Lossy compression","score":0.6022999882698059},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5619999766349792},{"id":"https://openalex.org/keywords/bernoullis-principle","display_name":"Bernoulli's principle","score":0.491100013256073},{"id":"https://openalex.org/keywords/hamming-distance","display_name":"Hamming distance","score":0.474700003862381},{"id":"https://openalex.org/keywords/block","display_name":"Block (permutation group theory)","score":0.47209998965263367},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.4366999864578247},{"id":"https://openalex.org/keywords/data-compression","display_name":"Data compression","score":0.3833000063896179}],"concepts":[{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.6388000249862671},{"id":"https://openalex.org/C165021410","wikidata":"https://www.wikidata.org/wiki/Q55564","display_name":"Lossy compression","level":2,"score":0.6022999882698059},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5619999766349792},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5264999866485596},{"id":"https://openalex.org/C152361515","wikidata":"https://www.wikidata.org/wiki/Q181328","display_name":"Bernoulli's principle","level":2,"score":0.491100013256073},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.474700003862381},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.47209998965263367},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.46939998865127563},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.4366999864578247},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3882000148296356},{"id":"https://openalex.org/C78548338","wikidata":"https://www.wikidata.org/wiki/Q2493","display_name":"Data compression","level":2,"score":0.3833000063896179},{"id":"https://openalex.org/C157125643","wikidata":"https://www.wikidata.org/wiki/Q884707","display_name":"Block code","level":3,"score":0.3522999882698059},{"id":"https://openalex.org/C113709454","wikidata":"https://www.wikidata.org/wiki/Q602136","display_name":"Coding theory","level":2,"score":0.34610000252723694},{"id":"https://openalex.org/C519991488","wikidata":"https://www.wikidata.org/wiki/Q28865","display_name":"Python (programming language)","level":2,"score":0.34369999170303345},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3411000072956085},{"id":"https://openalex.org/C81081738","wikidata":"https://www.wikidata.org/wiki/Q55542","display_name":"Lossless compression","level":3,"score":0.31949999928474426},{"id":"https://openalex.org/C63361517","wikidata":"https://www.wikidata.org/wiki/Q5645805","display_name":"Hamming weight","level":5,"score":0.3084999918937683},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3059999942779541},{"id":"https://openalex.org/C43126263","wikidata":"https://www.wikidata.org/wiki/Q128751","display_name":"Source code","level":2,"score":0.2913999855518341},{"id":"https://openalex.org/C41431624","wikidata":"https://www.wikidata.org/wiki/Q1053357","display_name":"Block size","level":3,"score":0.2881999909877777},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.2858999967575073},{"id":"https://openalex.org/C52622258","wikidata":"https://www.wikidata.org/wiki/Q131222","display_name":"Information theory","level":2,"score":0.2759000062942505},{"id":"https://openalex.org/C180016635","wikidata":"https://www.wikidata.org/wiki/Q2712821","display_name":"Compression (physics)","level":2,"score":0.2623000144958496},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.2621999979019165}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:doi:10.48550/arxiv.2602.24243","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.48550/arxiv.2602.24243","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2602.24243","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:doi:10.48550/arxiv.2602.24243","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Lossy":[0],"data":[1],"compression":[2],"lies":[3],"at":[4,28,128],"the":[5,17,54,66,82,88,96,101,105,117,123],"heart":[6],"of":[7],"modern":[8],"communication":[9],"and":[10,85,140],"storage":[11],"systems.":[12],"Shannon's":[13],"rate-distortion":[14,51,68],"theory":[15,52],"provides":[16],"fundamental":[18],"limit":[19,103],"on":[20,50],"how":[21,95],"much":[22],"a":[23,29,47,58],"source":[24],"can":[25],"be":[26],"compressed":[27],"given":[30],"fidelity,":[31],"but":[32],"it":[33],"assumes":[34],"infinitely":[35],"long":[36],"block":[37,90,106,130],"lengths":[38],"that":[39,93],"are":[40],"never":[41],"realized":[42],"in":[43,113],"practice.":[44],"We":[45,64,132],"present":[46],"self-contained":[48],"tutorial":[49],"for":[53,126],"simplest":[55],"non-trivial":[56],"source:":[57],"Bernoulli$(p)$":[59],"sequence":[60],"with":[61,137],"Hamming":[62],"distortion.":[63],"derive":[65],"classical":[67],"function":[69],"$RD":[70],"=":[71],"Hp":[72],"-":[73],"HD$":[74],"from":[75],"first":[76],"principles,":[77],"illustrate":[78],"its":[79],"computation":[80],"via":[81],"Blahut-Arimoto":[83],"algorithm,":[84],"then":[86],"develop":[87],"finite":[89,129],"length":[91,107],"refinements":[92],"characterize":[94],"minimum":[97],"achievable":[98],"rate":[99],"approaches":[100],"Shannon":[102],"as":[104],"$n$":[108],"grows.":[109],"The":[110],"central":[111],"quantity":[112],"this":[114],"refinement":[115],"is":[116],"\\emph{rate-distortion":[118],"dispersion}":[119],"$V(D)$,":[120],"which":[121],"governs":[122],"$O(1/\\sqrt{n})$":[124],"penalty":[125],"operating":[127],"lengths.":[131],"accompany":[133],"all":[134],"theoretical":[135],"developments":[136],"numerical":[138],"examples":[139],"figures":[141],"generated":[142],"by":[143],"accompanying":[144],"Python":[145],"scripts.":[146]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2026-03-03T00:00:00"}