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Cyclic relative difference sets and circulant weighing matrices

Daniel M. Gordon

TL;DR

This paper addressing the existence of cyclic relative difference sets that lift nontrivial base difference sets extends the program of Lam and Pott by performing extended searches up to $k\le256$ and examining implications for circulant weighing matrices. It develops and applies a framework leveraging multipliers, w-multipliers, and intersection-number constraints to rule out many potential liftings, finding no new nontrivial liftings beyond the Singer-complement parameters. The work connects RDS liftings to circulant weighing matrices via known constructions, clarifying when circumstantial Diophantine conditions (e.g., $4k = x^2 + qy^2$) can occur and updating catalogs of CW instances derived from RDS. Overall, it narrows the landscape of feasible liftings and provides new tables and online resources for researchers studying relative difference sets and circulant weighing matrices.

Abstract

An $(m,n,k,λ)$-relative difference set is a lifting of a $(m,k,nλ)$-difference set. Lam gave a table of cyclic relative difference sets with $k \leq 50$ in 1977, all of which were liftings of $( \frac{q^d-1}{q-1},q^{d-1},q^{d-2}(q-1))$-difference sets, the parameters of complements of classical Singer difference sets. Pott found all cyclic liftings of these difference sets with $n$ odd and $k \leq 64$ in 1995. No other nontrivial difference sets are known with liftings to relative difference sets, and Pott ended his survey on relative difference sets asking whether there are any others. In this paper we extend these searches, and apply the results to the existence of circulant weighing matrices.

Cyclic relative difference sets and circulant weighing matrices

TL;DR

This paper addressing the existence of cyclic relative difference sets that lift nontrivial base difference sets extends the program of Lam and Pott by performing extended searches up to and examining implications for circulant weighing matrices. It develops and applies a framework leveraging multipliers, w-multipliers, and intersection-number constraints to rule out many potential liftings, finding no new nontrivial liftings beyond the Singer-complement parameters. The work connects RDS liftings to circulant weighing matrices via known constructions, clarifying when circumstantial Diophantine conditions (e.g., ) can occur and updating catalogs of CW instances derived from RDS. Overall, it narrows the landscape of feasible liftings and provides new tables and online resources for researchers studying relative difference sets and circulant weighing matrices.

Abstract

An -relative difference set is a lifting of a -difference set. Lam gave a table of cyclic relative difference sets with in 1977, all of which were liftings of -difference sets, the parameters of complements of classical Singer difference sets. Pott found all cyclic liftings of these difference sets with odd and in 1995. No other nontrivial difference sets are known with liftings to relative difference sets, and Pott ended his survey on relative difference sets asking whether there are any others. In this paper we extend these searches, and apply the results to the existence of circulant weighing matrices.
Paper Structure (4 sections, 13 theorems, 12 equations, 7 tables)

This paper contains 4 sections, 13 theorems, 12 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Nonexistence Theorems for Relative Difference Sets
  3. Multipliers
  4. Circulant weighing matrices

Key Result

Theorem 1

Let $R$ be an $(m,n,k,\lambda)$-RDS in $G$. If $U$ is a normal subgroup of $G$ of order $u$ contained in $N$, then there exists a $(m,n/u,k,\lambda u)$-RDS in $G/U$ relative to $N/U$. In particular, $G/N$ contains an $(m,k,\lambda n)$-difference set.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Lemma 10
  • ...and 4 more