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Goethals--Seidel difference families with symmetric or skew base blocks

Dragomir Z. Djokovic, Ilias S. Kotsireas

TL;DR

The first examples of so called good matrices, G-matrices and best matrices of order 43, and goodMatrices and G-Matrix of order 45 are constructed, and some errors are pointed out.

Abstract

We single out a class of difference families which is widely used in some constructions of Hadamard matrices and which we call Goethals--Seidel (GS) difference families. They consist of four subsets (base blocks) of a finite abelian group of order $v$, which can be used to construct Hadamard matrices via the well-known Goethals--Seidel array. We consider the special class of these families in cyclic groups, where each base block is either symmetric or skew. We omit the well-known case where all four blocks are symmetric. By extending previous computations by several authors, we complete the classification of GS-difference families of this type for odd $v<50$. In particular, we have constructed the first examples of so called good matrices, G-matrices and best matrices of order 43, and good matrices and G-matrices of order 45. We also point out some errors in one of the cited references.

Goethals--Seidel difference families with symmetric or skew base blocks

TL;DR

The first examples of so called good matrices, G-matrices and best matrices of order 43, and goodMatrices and G-Matrix of order 45 are constructed, and some errors are pointed out.

Abstract

We single out a class of difference families which is widely used in some constructions of Hadamard matrices and which we call Goethals--Seidel (GS) difference families. They consist of four subsets (base blocks) of a finite abelian group of order , which can be used to construct Hadamard matrices via the well-known Goethals--Seidel array. We consider the special class of these families in cyclic groups, where each base block is either symmetric or skew. We omit the well-known case where all four blocks are symmetric. By extending previous computations by several authors, we complete the classification of GS-difference families of this type for odd . In particular, we have constructed the first examples of so called good matrices, G-matrices and best matrices of order 43, and good matrices and G-matrices of order 45. We also point out some errors in one of the cited references.

Paper Structure

This paper contains 18 sections, 4 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. GS-parameter sets
  3. Elementary transformations and equivalence relation
  4. Algorithmic issues
  5. Data collection
  6. Matching
  7. Summary of computational results and conclusions
  8. Acknowledgements
  9. Appendix
  10. $v=33$
  11. $v=35$
  12. $v=37$
  13. $v=39$
  14. $v=41$
  15. $v=43$
  16. ...and 3 more sections

Key Result

Proposition 1

The GS-parameter sets $(v;k_1,k_2,k_3,k_4;\lambda)$ with $v/2\ge k_1\ge k_2\ge k_3\ge k_4\ge 0$ are parametrized by four-odd-square decompositions $4v=\sum_{i=1}^4 s_i^2$ when $v$ is odd, and by four-square decompositions $v=\sum_{i=1}^4 s_i^2$ when $v$ is even. In both cases we require that $0\le s

Theorems & Definitions (8)

  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Definition 1