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Forbidden multipliers in abelian difference sets

Niklas Miller

TL;DR

This work investigates when certain integers can be multipliers of abelian difference sets and derives strong non-existence results and bounds for the multiplier group. By combining a group-ring Fourier-analytic framework with a strengthened Menon-type multiplier theorem and a parity-based refinement, the authors identify a broad class of forbidden multipliers in groups of the form $G=C_{p^e}\times H$ and deduce sharp size bounds for $M(D)$. They further analyze sub-difference-set structure and provide concrete consequences for cyclic and non-cyclic cases, including Hadamard-exception scenarios. The results significantly narrow the landscape of feasible abelian difference sets and provide practical criteria to rule out open parameter sets in design theory and related applications.

Abstract

We make the observation that certain group automorphisms that fix a large subgroup of an abelian group cannot be multipliers in any non-trivial abelian difference sets, with the single exception of an involution that can be a multiplier in Hadamard difference sets, provided that the difference set contains a sub-difference set of the same type. We use this observation together with a multiplier theorem to rule out the existence of difference sets, and derive bounds for the numerical multiplier group of a difference set.

Forbidden multipliers in abelian difference sets

TL;DR

This work investigates when certain integers can be multipliers of abelian difference sets and derives strong non-existence results and bounds for the multiplier group. By combining a group-ring Fourier-analytic framework with a strengthened Menon-type multiplier theorem and a parity-based refinement, the authors identify a broad class of forbidden multipliers in groups of the form and deduce sharp size bounds for . They further analyze sub-difference-set structure and provide concrete consequences for cyclic and non-cyclic cases, including Hadamard-exception scenarios. The results significantly narrow the landscape of feasible abelian difference sets and provide practical criteria to rule out open parameter sets in design theory and related applications.

Abstract

We make the observation that certain group automorphisms that fix a large subgroup of an abelian group cannot be multipliers in any non-trivial abelian difference sets, with the single exception of an involution that can be a multiplier in Hadamard difference sets, provided that the difference set contains a sub-difference set of the same type. We use this observation together with a multiplier theorem to rule out the existence of difference sets, and derive bounds for the numerical multiplier group of a difference set.

Paper Structure

This paper contains 6 sections, 15 theorems, 30 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Forbidden multipliers and sub-difference sets
  4. Bound on the size of the multiplier group
  5. A multiplier theorem and non-existence of difference sets
  6. Proofs

Key Result

Theorem 1

Let $G=C_{p^e}\times H$ where $e\geq 2$ and $H$ is an abelian group whose Sylow $p$-subgroup has exponent less than $p^e$. Suppose that $t$ is an integer such that $t\equiv 1+p^{e-1}\pmod{p^{e}}$ and $t\equiv 1\pmod{\text{exp}(H)}$. Then $t$ is not a multiplier in any non-trivial $(v,k,\lambda)$ dif

Figures (1)

  • Figure 1: A $(64,28,12)$ Hadamard difference set in $C_{16}\times C_4$ with $t=9$ as a multiplier. Note that column $j+8 \pmod{16}$ is equal to column $j \pmod{16}$ for all odd $j$, and the complement of column $j\pmod{16}$ for all even $j$.

Theorems & Definitions (24)

  • Theorem 1
  • Lemma 1: Fourier inversion formula
  • Lemma 2
  • proof
  • Lemma 3: marcus1977number
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • ...and 14 more