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

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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Bibliographic Details
Published inDesigns, codes, and cryptography Vol. 92; no. 4; pp. 1041 - 1049
Main Authors Miezaki, Tsuyoshi, Munemasa, Akihiro
Format Journal Article
LanguageEnglish
Published New York Springer US 01.04.2024
Springer Nature B.V
Subjects
Coding and Information Theory
Combinatorial analysis
Computer Science
Cryptology
Discrete Mathematics in Computer Science
Hamming codes
Polynomials
Extended Hamming code
Primary 94B05
design
Jacobi polynomial
Combinatorial
Harmonic weight enumerator
Reed–Muller code
Secondary 05B05
Online AccessGet full text
ISSN0925-1022
1573-7586
DOI10.1007/s10623-023-01327-0

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Summary: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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ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-023-01327-0