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Jacobi polynomials and harmonic weight enumerators of the first-order Reed–Muller codes and the extended Hamming codes

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Abstract

In the present paper, we give harmonic weight enumerators and Jacobi polynomials for the first-order Reed–Muller codes and the extended Hamming codes. As a corollary, we show the nonexistence of combinatorial 4-designs in these codes.

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References

  1. Assmus E.F. Jr., Mattson H.F. Jr.: New \(5\)-designs. J. Comb. Theory 6, 122–151 (1969).

    Article  MathSciNet  Google Scholar 

  2. Bachoc C.: On harmonic weight enumerators of binary codes. Des. Codes Cryptogr. 18(1–3), 11–28 (1999).

    Article  MathSciNet  Google Scholar 

  3. Bonnecaze A., Sole P.: The extended binary quadratic residue code of length 42 holds a 3-design. J. Comb. Des. 29(8), 528–532 (2021).

    Article  MathSciNet  Google Scholar 

  4. Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Compd. 24, 235–265 (1997).

    Article  MathSciNet  Google Scholar 

  5. Delsarte P.: Hahn polynomials, discrete harmonics, and \(t\)-designs. SIAM J. Appl. Math. 34(1), 157–166 (1978).

    Article  MathSciNet  Google Scholar 

  6. Dillion J.F., Schatz J.R.: Block designs with the symmetric difference property. In: Proceedings of the NSA Mathematical Sciences Meetings (Ward R. L. Ed.), pp. 159–164 (1987).

  7. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. II. North-Holland Mathematical Library, vol. 16. North-Holland Publishing Co., Amsterdam (1977).

    Google Scholar 

  8. Miezaki T., Munemasa A., Nakasora H.: A note on Assmus–Mattson type theorems. Des. Codes Cryptogr. 89, 843–858 (2021).

    Article  MathSciNet  Google Scholar 

  9. Miezaki T., Nakasora H.: The support designs of the triply even binary codes of length \(48\). J. Comb. Des. 27, 673–681 (2019).

    Article  MathSciNet  Google Scholar 

  10. Miezaki T., Nakasora H.: On the Assmus–Mattson type theorem for Type I and even formally self-dual codes. J. Comb. Des. 31(7), 335–344 (2023).

    Article  MathSciNet  Google Scholar 

  11. Ozeki M.: On the notion of Jacobi polynomials for codes. Math. Proc. Camb. Philos. Soc. 121(1), 15–30 (1997).

    Article  MathSciNet  Google Scholar 

  12. Wolfram Research, Inc., Mathematica, Version 11.2, Champaign, IL (2017).

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Acknowledgements

The authors are supported by JSPS KAKENHI (20K03527, 22K03277).

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Authors and Affiliations

  1. Faculty of Science and Engineering, Waseda University, Tokyo, 169-8555, Japan

    Tsuyoshi Miezaki

  2. Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Akihiro Munemasa

Authors
  1. Tsuyoshi Miezaki
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  2. Akihiro Munemasa
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Correspondence to Tsuyoshi Miezaki.

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The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript.

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Communicated by V. D. Tonchev.

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Miezaki, T., Munemasa, A. Jacobi polynomials and harmonic weight enumerators of the first-order Reed–Muller codes and the extended Hamming codes. Des. Codes Cryptogr. 92, 1041–1049 (2024). https://doi.org/10.1007/s10623-023-01327-0

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  • DOI: https://doi.org/10.1007/s10623-023-01327-0

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