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

Tsuyoshi Miezaki, Akihiro Munemasa

TL;DR

The paper computes explicit Jacobi polynomials $J_{C,T}$ and $J_{C^ot,T}$ for the first-order Reed–Muller codes $RM(1,m)$ and their duals (extended Hamming codes $H_{2^m}$) and derives harmonic weight enumerators $w_{C,f}$ and $w_{C^ot,f}$ using Bachoc’s framework. By analyzing these polynomials—particularly through subspace restrictions and MacWilliams-type identities—it establishes precise conditions under which shells fail to form combinatorial $4$-designs. It provides two independent proofs of the nonexistence of $4$-designs in any shell, one via Jacobi polynomial differences and another via a harmonic function in $ ext{Harm}_4$ drawn from a $3$-$(8,4,1)$ design. Collectively, the results rule out $4$-designs in the shells of $RM(1,m)$ and $H_{2^m}$ and showcase the power of Jacobi polynomials and harmonic weight enumerators in design-theoretic analyses of classical codes.

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.

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

TL;DR

The paper computes explicit Jacobi polynomials and for the first-order Reed–Muller codes and their duals (extended Hamming codes ) and derives harmonic weight enumerators and using Bachoc’s framework. By analyzing these polynomials—particularly through subspace restrictions and MacWilliams-type identities—it establishes precise conditions under which shells fail to form combinatorial -designs. It provides two independent proofs of the nonexistence of -designs in any shell, one via Jacobi polynomial differences and another via a harmonic function in drawn from a - design. Collectively, the results rule out -designs in the shells of and and showcase the power of Jacobi polynomials and harmonic weight enumerators in design-theoretic analyses of classical codes.

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 -designs in these codes.
Paper Structure (8 sections, 9 theorems, 27 equations)

This paper contains 8 sections, 9 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Codes and combinatorial $t$-designs
  4. Jacobi polynomials
  5. Harmonic weight enumerators
  6. Proof of Theorem \ref{['thm:Jacobi']}
  7. Proof of Theorem \ref{['thm:main1']}
  8. Proof of Corollary \ref{['cor:main']}

Key Result

Theorem 1.1

Let $C=RM(1,m)$ and $T=\{0,u_1,u_2,u_3\}\in\binom{V}{4}$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1: assmus-mattson
  • Theorem 2.2: Ozeki
  • Theorem 2.3: Delsarte
  • Definition 2.4
  • Theorem 2.5: Bachoc
  • Lemma 3.1
  • proof
  • ...and 5 more