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Three new lengths for cyclic Legendre pairs

N. A. Balonin, D. Ž. Đoković

Abstract

There are 20 odd integers v less than 200 for which the existence of Legendre pairs of length v is undecided. The smallest among them is v=77. We have constructed Legendre pairs of lengths 91, 93 and 123 reducing the number of undecided cases to 17.

Three new lengths for cyclic Legendre pairs

Abstract

There are 20 odd integers v less than 200 for which the existence of Legendre pairs of length v is undecided. The smallest among them is v=77. We have constructed Legendre pairs of lengths 91, 93 and 123 reducing the number of undecided cases to 17.

Paper Structure

This paper contains 7 sections, 1 theorem, 20 equations.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Cyclic Legendre pairs of new lengths $v=91,93,123$
  4. Type 1 and type 2
  5. New Legendre DFs of type 2
  6. Equivalence of Legendre pairs
  7. Acknowledgements

Key Result

Proposition 1

An ordered pair of functions $(f,g):G\to\{+1,-1\}$ is a Legendre pair on $G$ if and only if $(G_f,G_g)$ is a difference family in $G$ with parameters $(v;k_1,k_2;\lambda)$ where $k_1=|G_f|$, $k_2=|G_g|$ and $\lambda=k_1+k_2-(v+1)/2$. In particular the existence of Legendre pairs on $G$ implies that

Theorems & Definitions (6)

  • Definition 1
  • Proposition 1
  • proof
  • Remark 1
  • Conjecture 1
  • Definition 2