Few-weight codes from trace codes over a local ring View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2017-10-24

AUTHORS

Minjia Shi, Liqin Qian, Patrick Solé

ABSTRACT

In this paper, few weights linear codes over the local ring R=Fp+uFp+vFp+uvFp,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R={\mathbb {F}}_p+u{\mathbb {F}}_p+v{\mathbb {F}}_p+uv{\mathbb {F}}_p,$$\end{document} with u2=v2=0,uv=vu,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^2=v^2=0, uv=vu,$$\end{document} are constructed by using the trace function defined on an extension ring of degree m of R. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gauss sums over finite fields. Two different defining sets are explored. Using a linear Gray map from R to Fp4,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_p^4,$$\end{document} we obtain several families of p-ary codes from trace codes of dimension 4m. For two different defining sets: when m is even, or m is odd and p≡3(mod4).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\equiv 3 ~(\mathrm{mod} ~4).$$\end{document} Thus we obtain two family of p-ary abelian two-weight codes, which are directly related to MacDonald codes. When m is even and under some special conditions, we obtain two classes of three-weight codes. In addition, we give the minimum distance of the dual code. Finally, applications of the p-ary image codes in secret sharing schemes are presented. More... »

PAGES

335-350

References to SciGraph publications

  • 2008-02-28. Ring geometries, two-weight codes, and strongly regular graphs in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2016-09-27. Two and three weight codes over Fp+uFp in CRYPTOGRAPHY AND COMMUNICATIONS
  • 2003-06-18. Covering and Secret Sharing with Linear Codes in DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00200-017-0345-8

    DOI

    http://dx.doi.org/10.1007/s00200-017-0345-8

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1092340940


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "National Mobile Communications Research Laboratory, Southeast University, 210096, Nanjing, China", 
              "id": "http://www.grid.ac/institutes/grid.263826.b", 
              "name": [
                "Key Laboratory of Intelligent Computing Signal Processing, Ministry of Education, Anhui University, No. 3 Feixi Road, 230039, Hefei, China", 
                "School of Mathematical Sciences, Anhui University, 230601, Hefei, China", 
                "National Mobile Communications Research Laboratory, Southeast University, 210096, Nanjing, China"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Shi", 
            "givenName": "Minjia", 
            "id": "sg:person.012012432235.16", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012012432235.16"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "School of Mathematical Sciences, Anhui University, 230601, Hefei, China", 
              "id": "http://www.grid.ac/institutes/grid.252245.6", 
              "name": [
                "School of Mathematical Sciences, Anhui University, 230601, Hefei, China"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Qian", 
            "givenName": "Liqin", 
            "id": "sg:person.010326552045.60", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010326552045.60"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "CNRS/LAGA, University of Paris 8, 2 rue de la libert\u00e9, 93, Saint-Denis, France", 
              "id": "http://www.grid.ac/institutes/grid.15878.33", 
              "name": [
                "CNRS/LAGA, University of Paris 8, 2 rue de la libert\u00e9, 93, Saint-Denis, France"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Sol\u00e9", 
            "givenName": "Patrick", 
            "id": "sg:person.012750235663.02", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012750235663.02"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/3-540-45066-1_2", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1014896515", 
              "https://doi.org/10.1007/3-540-45066-1_2"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s12095-016-0206-5", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1024952889", 
              "https://doi.org/10.1007/s12095-016-0206-5"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10623-007-9136-8", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1038465688", 
              "https://doi.org/10.1007/s10623-007-9136-8"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2017-10-24", 
        "datePublishedReg": "2017-10-24", 
        "description": "In this paper, few weights linear codes over the local ring R=Fp+uFp+vFp+uvFp,\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R={\\mathbb {F}}_p+u{\\mathbb {F}}_p+v{\\mathbb {F}}_p+uv{\\mathbb {F}}_p,$$\\end{document} with u2=v2=0,uv=vu,\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$u^2=v^2=0, uv=vu,$$\\end{document} are constructed by using the trace function defined on an extension ring of degree m of R. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gauss sums over finite fields. Two different defining sets are explored. Using a linear Gray map from R to Fp4,\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathbb {F}}_p^4,$$\\end{document} we obtain several families of p-ary codes from trace codes of dimension 4m. For two different defining sets: when m is even, or m is odd and p\u22613(mod4).\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p\\equiv 3 ~(\\mathrm{mod} ~4).$$\\end{document} Thus we obtain two family of p-ary abelian two-weight codes, which are directly related to MacDonald codes. When m is even and under some special conditions, we obtain two classes of three-weight codes. In addition, we give the minimum distance of the dual code. Finally, applications of the p-ary image codes in secret sharing schemes are presented.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/s00200-017-0345-8", 
        "inLanguage": "en", 
        "isAccessibleForFree": false, 
        "isPartOf": [
          {
            "id": "sg:journal.1136287", 
            "issn": [
              "0938-1279", 
              "1432-0622"
            ], 
            "name": "Applicable Algebra in Engineering, Communication and Computing", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "4", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "29"
          }
        ], 
        "keywords": [
          "trace codes", 
          "algebraic structure", 
          "abelian codes", 
          "finite field", 
          "two-weight codes", 
          "MacDonald codes", 
          "three-weight codes", 
          "local ring R", 
          "ring R", 
          "trace function", 
          "extension ring", 
          "Gauss sums", 
          "defining set", 
          "set", 
          "Gray map", 
          "ary codes", 
          "minimum distance", 
          "dual code", 
          "secret sharing scheme", 
          "weight codes", 
          "local ring", 
          "code", 
          "function", 
          "sum", 
          "maps", 
          "dimensions", 
          "special conditions", 
          "class", 
          "applications", 
          "image code", 
          "sharing scheme", 
          "scheme", 
          "weight", 
          "U2", 
          "ring", 
          "degree", 
          "structure", 
          "weight distribution", 
          "distribution", 
          "means", 
          "field", 
          "family", 
          "conditions", 
          "addition", 
          "distance", 
          "paper", 
          "UV", 
          "different defining sets", 
          "linear Gray map", 
          "ary image codes"
        ], 
        "name": "Few-weight codes from trace codes over a local ring", 
        "pagination": "335-350", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1092340940"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s00200-017-0345-8"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s00200-017-0345-8", 
          "https://app.dimensions.ai/details/publication/pub.1092340940"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-01-01T18:44", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20220101/entities/gbq_results/article/article_753.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/s00200-017-0345-8"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00200-017-0345-8'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00200-017-0345-8'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00200-017-0345-8'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00200-017-0345-8'


     

    This table displays all metadata directly associated to this object as RDF triples.

    142 TRIPLES      22 PREDICATES      78 URIs      67 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s00200-017-0345-8 schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author N45195ad4f6a84240b4ec04441dbb8bc6
    4 schema:citation sg:pub.10.1007/3-540-45066-1_2
    5 sg:pub.10.1007/s10623-007-9136-8
    6 sg:pub.10.1007/s12095-016-0206-5
    7 schema:datePublished 2017-10-24
    8 schema:datePublishedReg 2017-10-24
    9 schema:description In this paper, few weights linear codes over the local ring R=Fp+uFp+vFp+uvFp,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R={\mathbb {F}}_p+u{\mathbb {F}}_p+v{\mathbb {F}}_p+uv{\mathbb {F}}_p,$$\end{document} with u2=v2=0,uv=vu,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^2=v^2=0, uv=vu,$$\end{document} are constructed by using the trace function defined on an extension ring of degree m of R. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gauss sums over finite fields. Two different defining sets are explored. Using a linear Gray map from R to Fp4,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_p^4,$$\end{document} we obtain several families of p-ary codes from trace codes of dimension 4m. For two different defining sets: when m is even, or m is odd and p≡3(mod4).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\equiv 3 ~(\mathrm{mod} ~4).$$\end{document} Thus we obtain two family of p-ary abelian two-weight codes, which are directly related to MacDonald codes. When m is even and under some special conditions, we obtain two classes of three-weight codes. In addition, we give the minimum distance of the dual code. Finally, applications of the p-ary image codes in secret sharing schemes are presented.
    10 schema:genre article
    11 schema:inLanguage en
    12 schema:isAccessibleForFree false
    13 schema:isPartOf N7348f7484bf14ae989aadb6c2a10323e
    14 N94fc0255e79e45b29391eb64f5602faf
    15 sg:journal.1136287
    16 schema:keywords Gauss sums
    17 Gray map
    18 MacDonald codes
    19 U2
    20 UV
    21 abelian codes
    22 addition
    23 algebraic structure
    24 applications
    25 ary codes
    26 ary image codes
    27 class
    28 code
    29 conditions
    30 defining set
    31 degree
    32 different defining sets
    33 dimensions
    34 distance
    35 distribution
    36 dual code
    37 extension ring
    38 family
    39 field
    40 finite field
    41 function
    42 image code
    43 linear Gray map
    44 local ring
    45 local ring R
    46 maps
    47 means
    48 minimum distance
    49 paper
    50 ring
    51 ring R
    52 scheme
    53 secret sharing scheme
    54 set
    55 sharing scheme
    56 special conditions
    57 structure
    58 sum
    59 three-weight codes
    60 trace codes
    61 trace function
    62 two-weight codes
    63 weight
    64 weight codes
    65 weight distribution
    66 schema:name Few-weight codes from trace codes over a local ring
    67 schema:pagination 335-350
    68 schema:productId N1cd09370be094d0b9868eb15033ca207
    69 Nb3c7a9e18fef42bda799531a03ce0b1a
    70 schema:sameAs https://app.dimensions.ai/details/publication/pub.1092340940
    71 https://doi.org/10.1007/s00200-017-0345-8
    72 schema:sdDatePublished 2022-01-01T18:44
    73 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    74 schema:sdPublisher Ne05ac2a145a04a1db24ab631dd3ea482
    75 schema:url https://doi.org/10.1007/s00200-017-0345-8
    76 sgo:license sg:explorer/license/
    77 sgo:sdDataset articles
    78 rdf:type schema:ScholarlyArticle
    79 N1cd09370be094d0b9868eb15033ca207 schema:name doi
    80 schema:value 10.1007/s00200-017-0345-8
    81 rdf:type schema:PropertyValue
    82 N45195ad4f6a84240b4ec04441dbb8bc6 rdf:first sg:person.012012432235.16
    83 rdf:rest Nf82e2e4f51214eba8bb4b2a96e36edc5
    84 N7348f7484bf14ae989aadb6c2a10323e schema:volumeNumber 29
    85 rdf:type schema:PublicationVolume
    86 N94fc0255e79e45b29391eb64f5602faf schema:issueNumber 4
    87 rdf:type schema:PublicationIssue
    88 Nb3c7a9e18fef42bda799531a03ce0b1a schema:name dimensions_id
    89 schema:value pub.1092340940
    90 rdf:type schema:PropertyValue
    91 Nb6206e29745f47848e19a8f9fca078eb rdf:first sg:person.012750235663.02
    92 rdf:rest rdf:nil
    93 Ne05ac2a145a04a1db24ab631dd3ea482 schema:name Springer Nature - SN SciGraph project
    94 rdf:type schema:Organization
    95 Nf82e2e4f51214eba8bb4b2a96e36edc5 rdf:first sg:person.010326552045.60
    96 rdf:rest Nb6206e29745f47848e19a8f9fca078eb
    97 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    98 schema:name Mathematical Sciences
    99 rdf:type schema:DefinedTerm
    100 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    101 schema:name Pure Mathematics
    102 rdf:type schema:DefinedTerm
    103 sg:journal.1136287 schema:issn 0938-1279
    104 1432-0622
    105 schema:name Applicable Algebra in Engineering, Communication and Computing
    106 schema:publisher Springer Nature
    107 rdf:type schema:Periodical
    108 sg:person.010326552045.60 schema:affiliation grid-institutes:grid.252245.6
    109 schema:familyName Qian
    110 schema:givenName Liqin
    111 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010326552045.60
    112 rdf:type schema:Person
    113 sg:person.012012432235.16 schema:affiliation grid-institutes:grid.263826.b
    114 schema:familyName Shi
    115 schema:givenName Minjia
    116 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012012432235.16
    117 rdf:type schema:Person
    118 sg:person.012750235663.02 schema:affiliation grid-institutes:grid.15878.33
    119 schema:familyName Solé
    120 schema:givenName Patrick
    121 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012750235663.02
    122 rdf:type schema:Person
    123 sg:pub.10.1007/3-540-45066-1_2 schema:sameAs https://app.dimensions.ai/details/publication/pub.1014896515
    124 https://doi.org/10.1007/3-540-45066-1_2
    125 rdf:type schema:CreativeWork
    126 sg:pub.10.1007/s10623-007-9136-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1038465688
    127 https://doi.org/10.1007/s10623-007-9136-8
    128 rdf:type schema:CreativeWork
    129 sg:pub.10.1007/s12095-016-0206-5 schema:sameAs https://app.dimensions.ai/details/publication/pub.1024952889
    130 https://doi.org/10.1007/s12095-016-0206-5
    131 rdf:type schema:CreativeWork
    132 grid-institutes:grid.15878.33 schema:alternateName CNRS/LAGA, University of Paris 8, 2 rue de la liberté, 93, Saint-Denis, France
    133 schema:name CNRS/LAGA, University of Paris 8, 2 rue de la liberté, 93, Saint-Denis, France
    134 rdf:type schema:Organization
    135 grid-institutes:grid.252245.6 schema:alternateName School of Mathematical Sciences, Anhui University, 230601, Hefei, China
    136 schema:name School of Mathematical Sciences, Anhui University, 230601, Hefei, China
    137 rdf:type schema:Organization
    138 grid-institutes:grid.263826.b schema:alternateName National Mobile Communications Research Laboratory, Southeast University, 210096, Nanjing, China
    139 schema:name Key Laboratory of Intelligent Computing Signal Processing, Ministry of Education, Anhui University, No. 3 Feixi Road, 230039, Hefei, China
    140 National Mobile Communications Research Laboratory, Southeast University, 210096, Nanjing, China
    141 School of Mathematical Sciences, Anhui University, 230601, Hefei, China
    142 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...