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Infinite series of $3$-designs in the extended quadratic residue code

Madoka Awada

TL;DR

The paper addresses the problem of generating $3$-designs from extended quadratic residue codes over ${\mathbb F}_{r^2}$ beyond those implied by transitivity or the Assmus–Mattson theorem. It develops a harmonic-analytic framework using Jacobi polynomials and harmonic weight enumerators, along with duadic-code duality, to prove a main result: if $p$ is an odd prime with $p \equiv 1 \pmod{4}$ and $r$ is a nonresidue modulo $p$, then every nonempty shell $(\widetilde{Q}_{r^2,p+1})_\ell$ is a $3$-design; special cases $r=2$ or $3$ (with $p$ satisfying congruence conditions) are also established. The paper then provides explicit examples and an extensive infinite series of $(r,p)$ pairs yielding $3$-designs, showing these designs often do not follow from transitivity or AM theorems. This work expands the known sources of $3$-designs in QR-code-related structures and demonstrates a robust method to obtain large families of designs in algebraic coding-theoretic objects with limited symmetry. The results have potential implications for combinatorial design theory and its applications in coding theory and related areas.

Abstract

In this paper, we show infinite series of $3$-designs in the extended quadratic residue codes over $\mathbb {F}_{r^2}$ for a prime $r$.

Infinite series of $3$-designs in the extended quadratic residue code

TL;DR

The paper addresses the problem of generating -designs from extended quadratic residue codes over beyond those implied by transitivity or the Assmus–Mattson theorem. It develops a harmonic-analytic framework using Jacobi polynomials and harmonic weight enumerators, along with duadic-code duality, to prove a main result: if is an odd prime with and is a nonresidue modulo , then every nonempty shell is a -design; special cases or (with satisfying congruence conditions) are also established. The paper then provides explicit examples and an extensive infinite series of pairs yielding -designs, showing these designs often do not follow from transitivity or AM theorems. This work expands the known sources of -designs in QR-code-related structures and demonstrates a robust method to obtain large families of designs in algebraic coding-theoretic objects with limited symmetry. The results have potential implications for combinatorial design theory and its applications in coding theory and related areas.

Abstract

In this paper, we show infinite series of -designs in the extended quadratic residue codes over for a prime .
Paper Structure (10 sections, 10 theorems, 34 equations)

This paper contains 10 sections, 10 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Codes and $t$-designs
  4. Jacobi polynomials
  5. Harmonic weight enumerators
  6. Preliminary steps of proving Theorem \ref{['thm:newmain']}
  7. Proof of Theorem \ref{['thm:newmain']}
  8. $3$-designs in the extended quadratic residue codes over ${\mathbb F}_{r^2}$
  9. Examples
  10. Infinite series of $3$-designs in the extended quadratic residue code over ${\mathbb F}_{r^2}$

Key Result

Theorem 1.1

Let $p$ be an odd prime. Let $\widetilde{Q}_{{r^2},p+1}$ be the extended quadratic residue code of length $p+1$ over ${\mathbb F}_{r^2}$. If $p \equiv1 \pmod{4}$ and $r$ is not a quadratic residue modulo $p$, then, for $\ell\in {\mathbb N}$, ${(\widetilde{Q}_{{r^2},p+1})}_\ell$ is a $3$-design whene

Theorems & Definitions (23)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1: DMS
  • Lemma 2.2: HP
  • Remark 2.3
  • Example 2.4
  • Theorem 2.5: assmus-mattson, Tanabe
  • Remark 2.6
  • Theorem 2.7: AMMN
  • Definition 2.8
  • ...and 13 more