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Determining the Complete Weight Distributions of Some Families of Cyclic Codes

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Arithmetic of Finite Fields (WAIFI 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15176))

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Abstract

Obtaining the complete weight distributions for nonbinary codes is an even harder problem than obtaining their Hamming weight distributions. In fact, obtaining these distributions is a problem that usually involves the evaluation of sophisticated exponential sums, which leaves this problem open for most of the linear codes. In this work we present a method that uses the known complete weight distribution of a given cyclic code, to determine the complete weight distributions of other cyclic codes. In addition we also obtain the complete weight distributions for a particular kind of one- and two-weight irreducible cyclic codes, and use these distributions and the method, in order to determine the complete weight distributions of infinite families of cyclic codes. As an example, and as a particular instance of our results, we determine in a simple way the complete weight distribution for one of the two families of reducible cyclic codes studied by Bae, Li and Yue [Discrete Mathematics, 338 (2015) 2275-2287].

This manuscript is partially supported by PAPIIT-UNAM IN107423. The second author has also received research support from CONAHCyT, México.

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Authors and Affiliations

  1. Dirección General de Cómputo y de Tecnologías de Información y Comunicación, Universidad Nacional Autónoma de México, 04510, Ciudad de México, Mexico

    Gerardo Vega

  2. Posgrado en Ciencia e Ingeniería de la Computación, Universidad Nacional Autónoma de México, 04510, Ciudad de México, Mexico

    Félix Hernández

Authors
  1. Gerardo Vega
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  2. Félix Hernández
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Correspondence to Gerardo Vega .

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Editors and Affiliations

  1. KU Leuven, Leuven, Belgium

    Svetla Petkova-Nikova

  2. Carleton University, Ottawa, ON, Canada

    Daniel Panario

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Vega, G., Hernández, F. (2025). Determining the Complete Weight Distributions of Some Families of Cyclic Codes. In: Petkova-Nikova, S., Panario, D. (eds) Arithmetic of Finite Fields. WAIFI 2024. Lecture Notes in Computer Science, vol 15176. Springer, Cham. https://doi.org/10.1007/978-3-031-81824-0_1

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  • DOI: https://doi.org/10.1007/978-3-031-81824-0_1

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  • Online ISBN: 978-3-031-81824-0

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