A generalization of combinatorial designs and related codes

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• 作者：Jerod Michel ; Baokun Ding
• 关键词：Almost difference set ; t ; Design ; Cyclotomic class ; Linear code ; t ; Adesign
• 刊名：Designs, Codes and Cryptography
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：82
• 期：3
• 页码：511-529
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Coding and Information Theory; Data Structures, Cryptology and Information Theory; Data Encryption; Discrete Mathematics in Computer Science; Information and Communication, Circuits;
• 出版者：Springer US
• ISSN：1573-7586
• 卷排序：82

文摘

In this paper we study a certain generalization of combinatorial designs related to almost difference sets, namely the t-adesign, which was coined by Ding (Codes from difference sets, 2015). It is clear that 2-adesigns are partially balanced incomplete block designs 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 some to new almost difference families), as well as two constructions of 3-adesigns. We discuss basic properties of the incidence matrices and make an initial investigation into the codes which they generate. We find that many of the codes have good parameters in the sense they are optimal or have relatively high minimum distance.