A Generalization of Combinatorial Designs Related to Almost Difference Sets

TL;DR

This paper explores a broad generalization of combinatorial designs called t-adesigns, providing new constructions, properties, and their applications in coding theory, extending the understanding of almost difference sets.

Contribution

It introduces new constructions of 2-adesigns and 3-adesigns, expanding the class of known almost difference sets and families, with analysis of their properties and applications.

Findings

New constructions of 2-adesigns and 3-adesigns

Identification of new almost difference sets and families

Analysis of incidence matrices and related codes

Abstract

In this paper we study a certain generalization of combinatorial designs related to almost difference sets, namely the $t$-adesign, which was coined by Cunsheng Ding in 2015. It is clear that $2$-adesigns are a kind of partially balanced incomplete block design which naturally arise in many combinatorial and statistical problems. We discuss some of their basic properties and give several constructions of $2$-adesigns (some of which correspond to new almost difference sets, and others of which correspond to new almost difference families), as well as two constructions of $3$-adesigns. We also discuss some basic properties of their incidence matrices and codes.