Characteristic Sets of Fixed-Dimension Vector Linear Codes for Non-Multicast Networks
Vector linear solvability of non-multicast networks depends upon both the characteristic of the finite field and the dimension of the vector linear network code. In the literature, the dependency on the characteristic of the finite field and the dependency on the dimension have been studied separate...
Gespeichert in:
| Veröffentlicht in: | IEEE transactions on information theory Jg. 66; H. 12; S. 7408 - 7426 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
IEEE
01.12.2020
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Schlagworte: | |
| ISSN: | 0018-9448, 1557-9654 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Abstract | Vector linear solvability of non-multicast networks depends upon both the characteristic of the finite field and the dimension of the vector linear network code. In the literature, the dependency on the characteristic of the finite field and the dependency on the dimension have been studied separately. In this paper, we show the interdependency between the characteristic of the finite field and the dimension of the vector linear network code that achieves a vector linear network coding (VLNC) solution in non-multicast networks. For any given network <inline-formula> <tex-math notation="LaTeX">\mathcal {N} </tex-math></inline-formula>, we define <inline-formula> <tex-math notation="LaTeX">P(\mathcal {N},d) </tex-math></inline-formula> as the set of all characteristics of finite fields over which the network <inline-formula> <tex-math notation="LaTeX">\mathcal {N} </tex-math></inline-formula> has a <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula>-dimensional VLNC solution. To the best of our knowledge, for any network <inline-formula> <tex-math notation="LaTeX">\mathcal {N} </tex-math></inline-formula> shown in the literature, if <inline-formula> <tex-math notation="LaTeX">P(\mathcal {N},1) </tex-math></inline-formula> is non-empty, then <inline-formula> <tex-math notation="LaTeX">P(\mathcal {N},1) = P(\mathcal {N},d) </tex-math></inline-formula> for any positive integer <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula>. We show that, for any two non-empty sets of primes <inline-formula> <tex-math notation="LaTeX">P_{1} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">P_{2} </tex-math></inline-formula>, there exists a network <inline-formula> <tex-math notation="LaTeX">\mathcal {N} </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">P(\mathcal {N},1) = P_{1} </tex-math></inline-formula>, but <inline-formula> <tex-math notation="LaTeX">P(\mathcal {N},2) = \{P_{1},P_{2} \} </tex-math></inline-formula>. We also show that there are networks exhibiting a similar advantage (the existence of a VLNC solution over a larger set of characteristics) if the dimension is increased from 2 to 3. However, such behaviour is not universal, as there exist networks which admit a VLNC solution over a smaller set of characteristics of finite fields when the dimension is increased. Using the networks constructed in this paper, we further demonstrate that: (i) a network having an <inline-formula> <tex-math notation="LaTeX">m_{1} </tex-math></inline-formula>-dimensional VLNC solution over a finite field of some characteristic and an <inline-formula> <tex-math notation="LaTeX">m_{2} </tex-math></inline-formula>-dimensional VLNC solution over a finite field of some other characteristic may not have an <inline-formula> <tex-math notation="LaTeX">(m_{1} + m_{2}) </tex-math></inline-formula>-dimensional VLNC solution over any finite field; (ii) there exist a class of networks for which scalar linear network coding (SLNC) over non-commutative rings has some advantage over SLNC over finite fields: the least sized non-commutative ring over which each network in the class has an SLNC solution is significantly lesser in size than the least sized finite field over which it has an SLNC solution. |
|---|---|
| AbstractList | Vector linear solvability of non-multicast networks depends upon both the characteristic of the finite field and the dimension of the vector linear network code. In the literature, the dependency on the characteristic of the finite field and the dependency on the dimension have been studied separately. In this paper, we show the interdependency between the characteristic of the finite field and the dimension of the vector linear network code that achieves a vector linear network coding (VLNC) solution in non-multicast networks. For any given network [Formula Omitted], we define [Formula Omitted] as the set of all characteristics of finite fields over which the network [Formula Omitted] has a [Formula Omitted]-dimensional VLNC solution. To the best of our knowledge, for any network [Formula Omitted] shown in the literature, if [Formula Omitted] is non-empty, then [Formula Omitted] for any positive integer [Formula Omitted]. We show that, for any two non-empty sets of primes [Formula Omitted] and [Formula Omitted], there exists a network [Formula Omitted] such that [Formula Omitted], but [Formula Omitted]. We also show that there are networks exhibiting a similar advantage (the existence of a VLNC solution over a larger set of characteristics) if the dimension is increased from 2 to 3. However, such behaviour is not universal, as there exist networks which admit a VLNC solution over a smaller set of characteristics of finite fields when the dimension is increased. Using the networks constructed in this paper, we further demonstrate that: (i) a network having an [Formula Omitted]-dimensional VLNC solution over a finite field of some characteristic and an [Formula Omitted]-dimensional VLNC solution over a finite field of some other characteristic may not have an [Formula Omitted]-dimensional VLNC solution over any finite field; (ii) there exist a class of networks for which scalar linear network coding (SLNC) over non-commutative rings has some advantage over SLNC over finite fields: the least sized non-commutative ring over which each network in the class has an SLNC solution is significantly lesser in size than the least sized finite field over which it has an SLNC solution. Vector linear solvability of non-multicast networks depends upon both the characteristic of the finite field and the dimension of the vector linear network code. In the literature, the dependency on the characteristic of the finite field and the dependency on the dimension have been studied separately. In this paper, we show the interdependency between the characteristic of the finite field and the dimension of the vector linear network code that achieves a vector linear network coding (VLNC) solution in non-multicast networks. For any given network <inline-formula> <tex-math notation="LaTeX">\mathcal {N} </tex-math></inline-formula>, we define <inline-formula> <tex-math notation="LaTeX">P(\mathcal {N},d) </tex-math></inline-formula> as the set of all characteristics of finite fields over which the network <inline-formula> <tex-math notation="LaTeX">\mathcal {N} </tex-math></inline-formula> has a <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula>-dimensional VLNC solution. To the best of our knowledge, for any network <inline-formula> <tex-math notation="LaTeX">\mathcal {N} </tex-math></inline-formula> shown in the literature, if <inline-formula> <tex-math notation="LaTeX">P(\mathcal {N},1) </tex-math></inline-formula> is non-empty, then <inline-formula> <tex-math notation="LaTeX">P(\mathcal {N},1) = P(\mathcal {N},d) </tex-math></inline-formula> for any positive integer <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula>. We show that, for any two non-empty sets of primes <inline-formula> <tex-math notation="LaTeX">P_{1} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">P_{2} </tex-math></inline-formula>, there exists a network <inline-formula> <tex-math notation="LaTeX">\mathcal {N} </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">P(\mathcal {N},1) = P_{1} </tex-math></inline-formula>, but <inline-formula> <tex-math notation="LaTeX">P(\mathcal {N},2) = \{P_{1},P_{2} \} </tex-math></inline-formula>. We also show that there are networks exhibiting a similar advantage (the existence of a VLNC solution over a larger set of characteristics) if the dimension is increased from 2 to 3. However, such behaviour is not universal, as there exist networks which admit a VLNC solution over a smaller set of characteristics of finite fields when the dimension is increased. Using the networks constructed in this paper, we further demonstrate that: (i) a network having an <inline-formula> <tex-math notation="LaTeX">m_{1} </tex-math></inline-formula>-dimensional VLNC solution over a finite field of some characteristic and an <inline-formula> <tex-math notation="LaTeX">m_{2} </tex-math></inline-formula>-dimensional VLNC solution over a finite field of some other characteristic may not have an <inline-formula> <tex-math notation="LaTeX">(m_{1} + m_{2}) </tex-math></inline-formula>-dimensional VLNC solution over any finite field; (ii) there exist a class of networks for which scalar linear network coding (SLNC) over non-commutative rings has some advantage over SLNC over finite fields: the least sized non-commutative ring over which each network in the class has an SLNC solution is significantly lesser in size than the least sized finite field over which it has an SLNC solution. |
| Author | Das, Niladri Rai, Brijesh Kumar |
| Author_xml | – sequence: 1 givenname: Niladri orcidid: 0000-0002-9494-2845 surname: Das fullname: Das, Niladri email: niladribegins@gmail.com organization: Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati, Guwahati, India – sequence: 2 givenname: Brijesh Kumar orcidid: 0000-0001-9719-956X surname: Rai fullname: Rai, Brijesh Kumar email: brijesh.rai@gmail.com organization: Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati, Guwahati, India |
| BookMark | eNo9kDFPwzAQhS1UJNrCjsQSiTnl7NhJPKJAoVIpAy1r5DhnkdLGxXYF_HtctWJ6utN7d3rfiAx62yMh1xQmlIK8W86WEwYMJhlQDoKfkSEVokhlLviADAFomUrOywsy8n4dRy4oG5JV9aGc0gFd50OnkzcMPrEmmXY_2KYP3RZ739k-eUcdrEvmXY_KJZVt0ScmLha2T1_2mxhVPiQLDN_WffpLcm7UxuPVScdkNX1cVs_p_PVpVt3PU804D6lCzQCaAkShTKmzRjcFl7nRtDGiaGgUKrEwJTXIsla1IIuGg8pBIlMUszG5Pd7dOfu1Rx_qtd27Pr6sGY_Fs5zKLLrg6NLOeu_Q1DvXbZX7rSnUB3h1hFcf4NUneDFyc4x0iPhvl1SUWc6yP42ObQA |
| CODEN | IETTAW |
| Cites_doi | 10.1109/TIT.2018.2866244 10.1109/TIT.2008.920209 10.1109/TIT.2005.847712 10.1109/LCOMM.2019.2922655 10.1109/TIT.2002.807285 10.1109/TIT.2010.2094930 10.1109/TCOMM.2016.2613085 10.1109/TIT.2016.2618379 10.1109/TIT.2007.896862 10.1109/18.850663 10.23919/ISITA.2018.8664366 10.1109/TIT.2017.2697421 10.1109/TIT.2015.2473863 10.1109/TIT.2005.851744 10.1109/TIT.2011.2169532 10.1109/TIT.2016.2613988 10.1109/TIT.2010.2090227 10.1109/TIT.2017.2697422 10.1109/TIT.2006.881746 10.1109/LCOMM.2016.2583418 10.1109/TIT.2018.2797183 10.1109/TIT.2016.2555955 |
| ContentType | Journal Article |
| Copyright | Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2020 |
| Copyright_xml | – notice: Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2020 |
| DBID | 97E RIA RIE AAYXX CITATION 7SC 7SP 8FD JQ2 L7M L~C L~D |
| DOI | 10.1109/TIT.2020.3014054 |
| DatabaseName | IEEE All-Society Periodicals Package (ASPP) 2005–Present IEEE All-Society Periodicals Package (ASPP) 1998–Present IEEE Electronic Library (IEL) CrossRef Computer and Information Systems Abstracts Electronics & Communications Abstracts Technology Research Database ProQuest Computer Science Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional |
| DatabaseTitle | CrossRef Technology Research Database Computer and Information Systems Abstracts – Academic Electronics & Communications Abstracts ProQuest Computer Science Collection Computer and Information Systems Abstracts Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Professional |
| DatabaseTitleList | Technology Research Database |
| Database_xml | – sequence: 1 dbid: RIE name: IEEE Electronic Library (IEL) url: https://ieeexplore.ieee.org/ sourceTypes: Publisher |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering Computer Science |
| EISSN | 1557-9654 |
| EndPage | 7426 |
| ExternalDocumentID | 10_1109_TIT_2020_3014054 9158362 |
| Genre | orig-research |
| GroupedDBID | -~X .DC 0R~ 29I 3EH 4.4 5GY 5VS 6IK 97E AAJGR AARMG AASAJ AAWTH ABAZT ABFSI ABQJQ ABVLG ACGFO ACGFS ACGOD ACIWK AENEX AETEA AETIX AGQYO AGSQL AHBIQ AI. AIBXA AKJIK AKQYR ALLEH ALMA_UNASSIGNED_HOLDINGS ASUFR ATWAV BEFXN BFFAM BGNUA BKEBE BPEOZ CS3 DU5 E.L EBS EJD F5P HZ~ H~9 IAAWW IBMZZ ICLAB IDIHD IFIPE IFJZH IPLJI JAVBF LAI M43 MS~ O9- OCL P2P PQQKQ RIA RIE RNS RXW TAE TN5 VH1 VJK AAYXX CITATION 7SC 7SP 8FD JQ2 L7M L~C L~D |
| ID | FETCH-LOGICAL-c244t-aec200b7057af8c3bcb7496fc1bf57b11bf19e7f81fe23dad097b40a609e2a1e3 |
| IEDL.DBID | RIE |
| ISICitedReferencesCount | 0 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000594905600008&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0018-9448 |
| IngestDate | Sun Jun 29 12:17:23 EDT 2025 Sat Nov 29 03:31:44 EST 2025 Wed Aug 27 02:27:29 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 12 |
| Language | English |
| License | https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html https://doi.org/10.15223/policy-029 https://doi.org/10.15223/policy-037 |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c244t-aec200b7057af8c3bcb7496fc1bf57b11bf19e7f81fe23dad097b40a609e2a1e3 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0001-9719-956X 0000-0002-9494-2845 |
| PQID | 2465436193 |
| PQPubID | 36024 |
| PageCount | 19 |
| ParticipantIDs | proquest_journals_2465436193 crossref_primary_10_1109_TIT_2020_3014054 ieee_primary_9158362 |
| PublicationCentury | 2000 |
| PublicationDate | 2020-12-01 |
| PublicationDateYYYYMMDD | 2020-12-01 |
| PublicationDate_xml | – month: 12 year: 2020 text: 2020-12-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | New York |
| PublicationPlace_xml | – name: New York |
| PublicationTitle | IEEE transactions on information theory |
| PublicationTitleAbbrev | TIT |
| PublicationYear | 2020 |
| Publisher | IEEE The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Publisher_xml | – name: IEEE – name: The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| References | ref13 ref24 ref23 ref15 ref14 ref20 ref11 ref22 ref10 ref21 ref2 ref1 ref17 ref16 ref19 ref18 ref8 ref7 ref4 ref3 jaggi (ref9) 2003 ref6 ref5 médard (ref12) 2003 |
| References_xml | – ident: ref23 doi: 10.1109/TIT.2018.2866244 – ident: ref18 doi: 10.1109/TIT.2008.920209 – start-page: 21 year: 2003 ident: ref12 article-title: On coding for non-multicast networks publication-title: Proc 41st Annu Allerton Conf Commun Control Comput – ident: ref4 doi: 10.1109/TIT.2005.847712 – ident: ref15 doi: 10.1109/LCOMM.2019.2922655 – ident: ref3 doi: 10.1109/TIT.2002.807285 – ident: ref8 doi: 10.1109/TIT.2010.2094930 – ident: ref10 doi: 10.1109/TCOMM.2016.2613085 – ident: ref17 doi: 10.1109/TIT.2016.2618379 – ident: ref13 doi: 10.1109/TIT.2007.896862 – ident: ref2 doi: 10.1109/18.850663 – ident: ref1 doi: 10.23919/ISITA.2018.8664366 – ident: ref21 doi: 10.1109/TIT.2017.2697421 – ident: ref6 doi: 10.1109/TIT.2015.2473863 – ident: ref16 doi: 10.1109/TIT.2005.851744 – ident: ref19 doi: 10.1109/TIT.2011.2169532 – ident: ref7 doi: 10.1109/TIT.2016.2613988 – ident: ref20 doi: 10.1109/TIT.2010.2090227 – ident: ref22 doi: 10.1109/TIT.2017.2697422 – ident: ref5 doi: 10.1109/TIT.2006.881746 – start-page: 1 year: 2003 ident: ref9 article-title: On linear network coding publication-title: Proc 42nd Annu Allerton Conf Commun Control Comput – ident: ref14 doi: 10.1109/LCOMM.2016.2583418 – ident: ref11 doi: 10.1109/TIT.2018.2797183 – ident: ref24 doi: 10.1109/TIT.2016.2555955 |
| SSID | ssj0014512 |
| Score | 2.3422465 |
| Snippet | Vector linear solvability of non-multicast networks depends upon both the characteristic of the finite field and the dimension of the vector linear network... |
| SourceID | proquest crossref ieee |
| SourceType | Aggregation Database Index Database Publisher |
| StartPage | 7408 |
| SubjectTerms | characteristic set Coding Commutativity Dependence Fields (mathematics) Knowledge engineering Linear codes M-network message dimension Modules (abstract algebra) Multicasting Network coding Networks non-multicast networks Numbers Rings (mathematics) Routing Sun Vector linear network coding vector linear solvability |
| Title | Characteristic Sets of Fixed-Dimension Vector Linear Codes for Non-Multicast Networks |
| URI | https://ieeexplore.ieee.org/document/9158362 https://www.proquest.com/docview/2465436193 |
| Volume | 66 |
| WOSCitedRecordID | wos000594905600008&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVIEE databaseName: IEEE Electronic Library (IEL) customDbUrl: eissn: 1557-9654 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0014512 issn: 0018-9448 databaseCode: RIE dateStart: 19630101 isFulltext: true titleUrlDefault: https://ieeexplore.ieee.org/ providerName: IEEE |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1NS8NAEB1q8aAHq61itcoevAhGs8k2mz1KtShIEaziLexuJuClERPFn-_sJi2KXjwlhySEnZ2defPxBuCEY4rCGIIlIpaBSEIVkFdtAmNRYZhaTHLPM3snZ7P0-Vndd-Bs1QuDiL74DM_drc_l56V9d6GyC8XHqT9w16RMml6tVcZAjHnDDM5JgQlzLFOSobqY384JCEaETx2cGIsfJsjPVPl1EHvrMu3977-2Yav1ItllI_Yd6OCiD73lhAbWKmwfNr_RDQ7gcfKDnZk9YF2xsmDTl0_MgyvH8-9iZ-zJR_IZwVRSAzYpc6wY-bZsVi4C37BrdVWzWVNAXu3C4_R6PrkJ2rEKgSVbXgcaLamGkeSp6SK1sbFGCpUUlptiLA2nC1coi5QXGMW5zkMljQg1yREjzTHeg-6iXOA-MBPGRYQ8KeJUC6MEmTqjNHltWhujtR3C6XKls9eGPSPzqCNUGUklc1LJWqkMYeBWdvVcu6hDGC1Fk7XqVWWRY4GLCfvFB3-_dQgb7ttN3ckIuvXbOx7Buv2oX6q3Y79zvgBzksJw |
| linkProvider | IEEE |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1LS8NAEB5EBfXgW6xW3YMXwWg22TbZo1SLYg2CrXgLu5sJ9NJIk4o_39lNWhS9eEoOCQk7OzvzzeMbgHOOMQqtCZaIMPJE15ceedXa0wYl-rHBbuZ4ZgdRksRvb_J5CS4XvTCI6IrP8Mreulx-VpiZDZVdS96J3YG7YidnNd1ai5yB6PCaG5yTChPqmCclfXk9fBgSFAwIoVpA0RE_jJCbqvLrKHb2pb_1vz_bhs3Gj2Q3teB3YAknu7A1n9HAGpXdhY1vhIN7MOr94GdmL1iVrMhZf_yJmXdrmf5t9Iy9ulg-I6BKisB6RYYlI--WJcXEcy27RpUVS-oS8nIfRv27Ye_eawYreIaseeUpNKQcOiJfTeWxCbXRkZDd3HCddyLN6cIlRnnMcwzCTGW-jLTwFUkSA8UxPIDlSTHBQ2DaD_MAeTcPYyW0FGTstFTktymltVKmBRfzlU7fa_6M1OEOX6YkldRKJW2k0oI9u7KL55pFbUF7Lpq0UbAyDSwPXEjoLzz6-60zWLsfPg3SwUPyeAzr9jt1FUoblqvpDE9g1XxU43J66nbRF7-7xbk |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Characteristic+Sets+of+Fixed-Dimension+Vector+Linear+Codes+for+Non-Multicast+Networks&rft.jtitle=IEEE+transactions+on+information+theory&rft.au=Das%2C+Niladri&rft.au=Rai%2C+Brijesh+Kumar&rft.date=2020-12-01&rft.pub=The+Institute+of+Electrical+and+Electronics+Engineers%2C+Inc.+%28IEEE%29&rft.issn=0018-9448&rft.eissn=1557-9654&rft.volume=66&rft.issue=12&rft.spage=7408&rft_id=info:doi/10.1109%2FTIT.2020.3014054&rft.externalDBID=NO_FULL_TEXT |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0018-9448&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0018-9448&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0018-9448&client=summon |