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Spread Construction for (36,15,6) Hadamard Difference Sets

Ken Smith, Jordan Webster

Abstract

There are exactly 35 inequivalent (36, 15, 6) difference sets in nine groups. Eight of the nine groups have a normal Sylow 3-subgroup. We give a straightforward spread construction which explains the 32 inequivalent difference sets in these eight groups. An interesting variation on this construction provides the three difference sets in the ninth group.

Spread Construction for (36,15,6) Hadamard Difference Sets

Abstract

There are exactly 35 inequivalent (36, 15, 6) difference sets in nine groups. Eight of the nine groups have a normal Sylow 3-subgroup. We give a straightforward spread construction which explains the 32 inequivalent difference sets in these eight groups. An interesting variation on this construction provides the three difference sets in the ninth group.
Paper Structure (7 sections, 5 theorems, 14 equations)

This paper contains 7 sections, 5 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Product constructions
  3. The Spread Construction
  4. Using Relative Difference Sets
  5. The nonlinear irreducible representations
  6. The linear representations of $G$
  7. Conclusions

Key Result

Theorem 1

If $G$, $H$, and $K$ are groups such that $G=H\times K$ and $D_{K}$ is a difference set in $K$ and $D_{H}$ is a difference set in $H$, then is the Hadamard transform of a difference set.

Theorems & Definitions (6)

  • Theorem 1: Menon
  • Theorem 2: Dillon
  • Theorem 3: The Spread Construction
  • Lemma 1
  • Proposition 1
  • proof