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Binary linear codes with at most three weights from cyclotomic mappings.

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Title: Binary linear codes with at most three weights from cyclotomic mappings.
Authors: Cui, Heming1 (AUTHOR) s23090029@s.upc.edu.cn, Zhao, Xubo1 (AUTHOR) zhaoxubo@upc.edu.cn, Wang, Qiang2 (AUTHOR) wang@math.carleton.ca, Li, Xiaoping1 (AUTHOR) xplteaa@163.com, Yan, Tongjiang1,3 (AUTHOR) yantoji@163.com
Source: Finite Fields & Their Applications. Jun2026, Vol. 112, pN.PAG-N.PAG. 1p.
Subjects: Linear codes, Cyclotomic fields, Message authentication codes, Cryptography
Abstract: Linear codes with few weights have attracted significant interest due to their wide-ranging applications in secret sharing, authentication codes, association schemes, strongly regular graph, and some other fields. This paper focuses on unifying several existing construction methods for few-weight linear codes, extending the works of Wang et al. (2015) [24] , Wu et al. (2019) [25] , and Fang et al. (2023) [10]. In our code construction, we introduce a novel index set J , whose cardinality and structural properties are shown to critically influence both the length and weight distribution of the resulting few-weight linear codes. By employing cyclotomic mappings and choosing the more general defining sets, several new classes of binary linear codes with at most three weights are constructed. Our framework subsumes all aforementioned constructions as special cases and enlarges the spectrum of attainable parameters. The weight distributions of the corresponding linear codes are also explicitly determined. We also demonstrate that some of the linear codes constructed in this paper are optimal in the sense that they have the best known parameters in the tables maintained by Markus Grassl and/or optimal in the sense that they meet certain bounds on linear codes. [ABSTRACT FROM AUTHOR]
Copyright of Finite Fields & Their Applications is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 192152324
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  Data: Binary linear codes with at most three weights from cyclotomic mappings.
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  Data: <searchLink fieldCode="AR" term="%22Cui%2C+Heming%22">Cui, Heming</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> s23090029@s.upc.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Zhao%2C+Xubo%22">Zhao, Xubo</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> zhaoxubo@upc.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Wang%2C+Qiang%22">Wang, Qiang</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> wang@math.carleton.ca</i><br /><searchLink fieldCode="AR" term="%22Li%2C+Xiaoping%22">Li, Xiaoping</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> xplteaa@163.com</i><br /><searchLink fieldCode="AR" term="%22Yan%2C+Tongjiang%22">Yan, Tongjiang</searchLink><relatesTo>1,3</relatesTo> (AUTHOR)<i> yantoji@163.com</i>
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  Data: <searchLink fieldCode="JN" term="%22Finite+Fields+%26+Their+Applications%22">Finite Fields & Their Applications</searchLink>. Jun2026, Vol. 112, pN.PAG-N.PAG. 1p.
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  Data: <searchLink fieldCode="DE" term="%22Linear+codes%22">Linear codes</searchLink><br /><searchLink fieldCode="DE" term="%22Cyclotomic+fields%22">Cyclotomic fields</searchLink><br /><searchLink fieldCode="DE" term="%22Message+authentication+codes%22">Message authentication codes</searchLink><br /><searchLink fieldCode="DE" term="%22Cryptography%22">Cryptography</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Linear codes with few weights have attracted significant interest due to their wide-ranging applications in secret sharing, authentication codes, association schemes, strongly regular graph, and some other fields. This paper focuses on unifying several existing construction methods for few-weight linear codes, extending the works of Wang et al. (2015) [24] , Wu et al. (2019) [25] , and Fang et al. (2023) [10]. In our code construction, we introduce a novel index set J , whose cardinality and structural properties are shown to critically influence both the length and weight distribution of the resulting few-weight linear codes. By employing cyclotomic mappings and choosing the more general defining sets, several new classes of binary linear codes with at most three weights are constructed. Our framework subsumes all aforementioned constructions as special cases and enlarges the spectrum of attainable parameters. The weight distributions of the corresponding linear codes are also explicitly determined. We also demonstrate that some of the linear codes constructed in this paper are optimal in the sense that they have the best known parameters in the tables maintained by Markus Grassl and/or optimal in the sense that they meet certain bounds on linear codes. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Finite Fields & Their Applications is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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    Identifiers:
      – Type: doi
        Value: 10.1016/j.ffa.2026.102798
    Languages:
      – Code: eng
        Text: English
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      Pagination:
        PageCount: 1
        StartPage: N.PAG
    Subjects:
      – SubjectFull: Linear codes
        Type: general
      – SubjectFull: Cyclotomic fields
        Type: general
      – SubjectFull: Message authentication codes
        Type: general
      – SubjectFull: Cryptography
        Type: general
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      – TitleFull: Binary linear codes with at most three weights from cyclotomic mappings.
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            NameFull: Cui, Heming
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            NameFull: Zhao, Xubo
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            NameFull: Wang, Qiang
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            NameFull: Li, Xiaoping
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            NameFull: Yan, Tongjiang
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          Dates:
            – D: 01
              M: 06
              Text: Jun2026
              Type: published
              Y: 2026
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              Value: 112
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