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Designs, Codes and Cryptography

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10-04-2023

Signed difference sets

Author: Daniel M. Gordon

Published in: Designs, Codes and Cryptography | Issue 5/2023

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Abstract

A $(v,k,\lambda )$ difference set in a group G is a subset $\{d_1, d_2, \ldots ,d_k\}$ of G such that $D=\sum d_i$ in the group ring ${\mathbb {Z}}[G]$ satisfies

$$\begin{aligned} D D^{-1} = n + \lambda G, \end{aligned}$$

where $n=k-\lambda $. If $D=\sum s_i d_i$, where the $s_i \in \{ \pm 1\}$, satisfies the same equation, we will call it a signed difference set. This generalizes both difference sets (all $s_i=1$) and circulant weighing matrices (G cyclic and $\lambda =0$). We will show that there are other cases of interest, and give some results on their existence.

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Title: Signed difference sets
Author: Daniel M. Gordon
Publication date: 10-04-2023
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 5/2023
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-022-01171-8

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