AI 中文总结
本文对凯莱码构造提出两种推广（图码与有向图码），分析所用（有向）图扩展性质与码参数的关系，改进了相关结果，构造出无限好有向图码族并提出开放问题。
AI 中文摘要
2012年，考夫曼（Kaufman）和卢博茨基（Lubotzky）构造了首个对称LDPC好码族，其构造使用了考夫曼和维格德森（Kaufman and Wigderson，2016）最初定义的凯莱码。本文对凯莱码构造提出两种推广，分别称为图码和有向图码。我们研究这些构造的代数与组合性质，证明它们具备凯莱码的相同优良属性，同时拥有更多自由度。我们分析了所用（有向）图的扩展性质与所构造码参数之间的关系，该分析改进了考夫曼和卢博茨基的结果。作为应用，我们构造了一个无限的好有向图码族，并提出一系列开放问题。
英文摘要
In 2012, Kaufman and Lubotzky constructed the first family of symmetric LDPC good codes. Their construction used Cayley codes, as originally defined by Kaufman and Wigderson (2016). In this paper we present two generalisations to the Cayley code construction, which we call graph codes and digraph codes. We investigate both the algebraic, and combinatorial properties of these constructions and show that they possess the same desirable attributes as Cayley codes, but with added freedom. We analyse the relationship between the expansion properties of the ingredient (di)graphs and the parameters of the constructed codes; our analysis offers an improvement to the results of Kaufman and Lubotzky. As an application, we construct an infinite family of good digraph codes, and we propose a series of open problems.
Comments18 pages, 1 figure
