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A note on $t$-designs in isodual codes

Madoka Awada, Tsuyoshi Miezaki, Akihiro Munemasa, Hiroyuki Nakasora

TL;DR

The paper addresses constructing combinatorial 3-designs from extended binary quadratic residue codes and their duals by combining Jacobi polynomials with harmonic weight enumerators. It shows that for primes with $p\equiv 1\pmod{8}$, the union of relevant shells in $\widetilde{Q}_{p+1}$ and its dual yields a 3-design when nonempty, leveraging isoduality and two Aut-orbits. The authors provide a detailed analysis of the length-42 case, proving that $(\widetilde{Q}_{42})_{10}$ forms a 3-(42,10,18) design and $(\widetilde{Q}_{42})_{32}$ forms a 3-(42,32,744) design, while other shells are not 3-designs; this extends known 2-designs via transitivity and supports new 3-designs beyond the standard Assmus--Mattson framework. The results enhance understanding of how algebraic- combinatorial symmetries in QR codes engender higher-design structures with potential applications in coding theory and combinatorics.

Abstract

In the present paper, we construct 3-designs using extended binary quadratic residue codes and their dual codes.

A note on $t$-designs in isodual codes

TL;DR

The paper addresses constructing combinatorial 3-designs from extended binary quadratic residue codes and their duals by combining Jacobi polynomials with harmonic weight enumerators. It shows that for primes with , the union of relevant shells in and its dual yields a 3-design when nonempty, leveraging isoduality and two Aut-orbits. The authors provide a detailed analysis of the length-42 case, proving that forms a 3-(42,10,18) design and forms a 3-(42,32,744) design, while other shells are not 3-designs; this extends known 2-designs via transitivity and supports new 3-designs beyond the standard Assmus--Mattson framework. The results enhance understanding of how algebraic- combinatorial symmetries in QR codes engender higher-design structures with potential applications in coding theory and combinatorics.

Abstract

In the present paper, we construct 3-designs using extended binary quadratic residue codes and their dual codes.
Paper Structure (7 sections, 6 theorems, 35 equations)

This paper contains 7 sections, 6 theorems, 35 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:main']}
  7. Quadratic residue code of length 42

Key Result

Theorem 1.1

Let $C$ be an isodual binary code of length $n$, $X:=\{1,\ldots,n\}$, and $G={\rm Aut}(C)$. Let $\sigma \in S_n$ such that $C^\perp=C^\sigma$. Then $G$ acts on $\binom{X}{t}$ and we assume that $G$ has two orbits: such that $(GT_1)^\sigma=GT_2$. Then the following statements hold:

Theorems & Definitions (18)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Example 2.1
  • Lemma 2.2: CL
  • Remark 2.3
  • Theorem 2.4: Delsarte
  • Theorem 2.5
  • proof
  • Definition 2.6
  • ...and 8 more