Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algorithms for difference families in finite abelian groups

Dragomir Z. Djokovic, Ilias S. Kotsireas

Abstract

Our main objective is to show that the computational methods that we previously developed to search for difference families in cyclic groups can be fully extended to the more general case of arbitrary finite abelian groups. In particular the power density PSD-test and the method of compression can be used to help the search.

Algorithms for difference families in finite abelian groups

Abstract

Our main objective is to show that the computational methods that we previously developed to search for difference families in cyclic groups can be fully extended to the more general case of arbitrary finite abelian groups. In particular the power density PSD-test and the method of compression can be used to help the search.

Paper Structure

This paper contains 13 sections, 9 theorems, 74 equations.

Table of Contents

  1. Difference families
  2. Difference families and the group algebra
  3. Characters of finite abelian groups
  4. Discrete Fourier transform, DFT
  5. Complementary functions
  6. PSD-test
  7. Compression of complementary functions
  8. Regular representation
  9. Some special classes of difference families
  10. DO-matrices and DO-difference families $(\alpha=2)$
  11. Periodic Golay pairs $(\alpha=0)$
  12. Legendre pairs $(\alpha=-2)$
  13. Goethals-Seidel quadruples $(\alpha=0)$

Key Result

Lemma 1

Let $(X_1,X_2,\ldots,X_t)$ be a $t$-tuple of proper nonempty subsets of $G$ with cardinalities $k_i=|X_i|$, and let $\lambda$ be a nonnegative integer. Then $(X_1,X_2,\ldots,X_t)$ is a difference family in $G$ with the parameter set $(v;k_1,k_2,\ldots,k_t;\lambda)$ if and only if where $n$ is defined by (eq:par-n).

Theorems & Definitions (19)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 9 more