nach oben

Designs, Codes and Cryptography

Erschienen in:

16.11.2023

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

verfasst von: Tsuyoshi Miezaki, Akihiro Munemasa

Erschienen in: Designs, Codes and Cryptography | Ausgabe 4/2024

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Vorheriger Artikel Improved bounds for permutation arrays under Chebyshev distance

Nächster Artikel Jacobi sums over Galois rings of arbitrary characters and their applications in constructing asymptotically optimal codebooks

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

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

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).MathSciNetCrossRef

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

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

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).

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).

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

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

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).MathSciNetCrossRef

11.

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

12.

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

Titel: Jacobi polynomials and harmonic weight enumerators of the first-order Reed–Muller codes and the extended Hamming codes
verfasst von: Tsuyoshi Miezaki
Akihiro Munemasa
Publikationsdatum: 16.11.2023
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 4/2024
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-023-01327-0

Springer Professional

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

Abstract

Premium Partner

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Weitere Artikel der Ausgabe 4/2024

Unconditionally secure non-malleable secret sharing and circular external difference families

Lattice codes for lattice-based PKE

LowMS: a new rank metric code-based KEM without ideal structure

Improved bounds for permutation arrays under Chebyshev distance

Hulls of linear codes from simplex codes

Jacobi sums over Galois rings of arbitrary characters and their applications in constructing asymptotically optimal codebooks

Premium Partner