最优多元常重码的构造

Constant-weight codes have important applications in many fields, including bandwidth-efficient channels, optical code-division multiple-access systems, mobile radar, frequency-hopping spread-spectrum communications and sonar signal designs, and are a research hotspot in combinatorics and coding theory. The constructions of constant-weight codes with more codewords and the improvements of their upper and lower bounds are the core topics in the research area related to constant-weight codes. Aiming at the problems of upper and lower bounds and constructions of q-ary constant-weight codes (briefly CWCs) and q-ary cyclic CWCs, this project will comprehensively explore the methods of combinatorial design theory, algebra, computer search algorithms and linear programming to improve the known upper and lower bounds of these two classes of codes and establish their new constructions. The specific works will include: (1) The study of the upper and lower bounds of binary CWCs by employing algebraic methods and linear programming approaches; (2) The research of the constructions of optimal q-ary CWCs with q>2; (3) The study of upper bounds, optimal direct and recursive constructions of q-ary cyclic CWCs. This project will solve some important science problems related to these two classes of codes, and provide some novel approaches for their upper-lower bound analysis and constructions. Further it can help enrich and extend the theory and methods of combinatorial designs.

常重码在高效带宽信道、光码分多址系统、移动雷达、跳频扩频通信和声纳信号设计等诸多领域有着重要的应用，是组合设计和编码理论的研究热点。构造码字容量更大的常重码和改进常重码的已有上下界是常重码研究的核心课题。本项目针对多元常重码和多元循环常重码的上下界问题和构造问题，综合运用代数方法、组合设计理论、计算机搜索算法和线性规划等工具，改进这两类码的已有上下界，建立它们的新优构造方法。本项目的主要工作包括：（1）利用代数方法和线性规划方法，研究二元常重码的上下界；（2）研究最优q元常重码的构造方法，其中q>2；（2）研究多元循环常重码的上界、最优直接构造和递推构造。本项目预期解决这两类码相关的一些重要科学问题，为其上下界分析和构造提供新方法和新途径，同时丰富和扩展组合设计的理论和方法。

常重码在通信领域有着诸多重要应用，而几何正交码在DNA纳米材料的合成中起着重要作用。本项目旨在利用组合设计的理论和研究方法，讨论常重码、循环常重码和几何正交码的构造问题，主要内容包括：讨论多元循环常重码的上界问题及构造问题；探索几何正交码的构造问题；研究常重码的搜索算法，讨论搜索算法在用户画像、恶意电子邮件拦截等领域的应用。本项目丰富了组合学的内容，也有助于组合学各分支学科之间、与其他学科的交叉和渗透。