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Jacobi polynomials for the first-order generalized Reed--Muller codes

Ryosuke Yamaguchi

Abstract

In this paper, we give the Jacobi polynomials for first-order generalized Reed--Muller codes. We show as a corollary the nonexistence of combinatorial $3$-designs in these codes.

Jacobi polynomials for the first-order generalized Reed--Muller codes

Abstract

In this paper, we give the Jacobi polynomials for first-order generalized Reed--Muller codes. We show as a corollary the nonexistence of combinatorial -designs in these codes.
Paper Structure (8 sections, 17 theorems, 53 equations)

This paper contains 8 sections, 17 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Codes and combinatorial $t$-designs.
  4. Jacobi poynomials.
  5. Notation
  6. Proofs of Theorem \ref{['thm:2-design']} and Corollary \ref{['cor:2-design']}
  7. Proofs of Theorem \ref{['thm:3-design']} and Corollary \ref{['cor:3-design']}
  8. Proof of Theorem \ref{['thm:4-design']}

Key Result

Theorem 1.1

Let $C=RM_q(1, m)$ and $T = \{0, u\} \in \binom{V}{2}$. Then,

Theorems & Definitions (33)

  • Remark
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 2.1: Theorem 4 of Ozeki
  • Proposition 2.2
  • ...and 23 more