有向可分组 3-设计及其应用

项目名称： 有向可分组 3-设计及其应用

项目编号： No.11471096

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 田子红

作者单位： 河北师范大学

项目金额： 65万元

中文摘要： 组合设计是组合数学的一个重要分支，主要研究各种离散结构的存在性、构造和分类等问题。近年来，有着应用背景的设计存在问题和经典的组合设计存在问题一直是组合设计领域国内外研究的热点。本项目拟讨论几类有向3-设计的存在性及其在编码密码学中的应用。研究内容主要包括：区组长度为3, 4，组型一致的有向可分组3-设计和有向烛台形系，以及其在完备删位和插位纠错码、常重复合码等方面的应用；带有可分解性质、可划分性质的Mendelsohn型可分组3-设计和Mendelsohn烛台形系，以及其在构作特殊类型的Mendelsohn三元系大集方面的应用。这些课题的研究具有重要的理论意义和应用价值。

中文关键词： 有向设计；有向可分组3-设计；有向烛台形系；删位和插位纠错码；常重复合码

英文摘要： Combinatorial design theory is one of the important branches of combinatorial mathematics, which mainly studies the existence, construction and classification problems of discrete configurations. In recent years, the hot research areas of combinatorial design theory are the problems with application backgrounds and classical designs. This project will investigate the existence of several directed 3-designs and their applications in coding theory and cryptography. We will study uniform directed group divisible 3-designs and uniform directed candelabra systems with block sizes three or four，which will be used to construct perfect deletion and insertion-correcting codes, constant composition codes and so on; Mendelsohn group divisible 3-designs and Mendelsohn candelabra systems with resolvable or partitioned property, which will be used to construct the large sets of Mendelsohn triple systems with special types. The research on these subjects has important theoretical and applicable value.

英文关键词： directed design;directed group divisible 3-design;directed candelabra system;deletion and insertion-correcting code;constant composition code