We introduce the first example of algebraically constructed hierarchical quasi-cyclic codes. These codes are built from Reed-Solomon codes using a 1964 construction of superimposed codes by Kautz and Singleton. We show both the number of levels in the hierarchy and the index of these Reed-Solomon derived codes are determined by the field size. We show that this property also holds for certain additional classes of polynomial evaluation codes. We provide explicit code parameters and properties as well as some additional bounds on parameters such as rank and distance. In particular, starting with Reed-Solomon codes of dimension $k=2$ yields hierarchical quasi-cyclic codes with Tanner graphs of girth 6. We present a table of small code parameters and note that some of these codes meet the best known minimum distance for binary codes, with the additional hierarchical quasi-cyclic structure. We draw connections to similar constructions in the literature, but importantly, while existing literature on related codes is largely simulation-based, we present a novel algebraic approach to determining new bounds on parameters of these codes.

翻译：本文首次提出了代数构造的分层准循环码。这些码基于里德-所罗门码，采用Kautz与Singleton于1964年提出的叠加码构造方法构建。我们证明了该层次结构的级数以及这些由里德-所罗门码导出的码的指数均由域的大小决定。我们还证明了该性质对某些其他类型的多项式求值码同样成立。我们给出了明确的码参数与性质，并对秩、距离等参数提供了额外的界。特别地，从维度$k=2$的里德-所罗门码出发，可构造出Tanner图围长为6的分层准循环码。我们列出了小参数码表，并指出其中部分码在具备分层准循环结构的同时，达到了二元码当前已知的最佳最小距离。我们将其与文献中的类似构造进行了关联比较，但重要的是，现有相关码的研究主要基于仿真，而本文提出了一种新颖的代数方法来确定这些码参数的新界。