Hierarchical Quasi-cyclic Codes from Reed-Solomon and Polynomial Evaluation Codes

TL;DR

This paper introduces a novel algebraic construction of hierarchical quasi-cyclic codes derived from Reed-Solomon and polynomial evaluation codes, providing explicit parameters, bounds, and demonstrating their optimality in certain cases.

Contribution

It presents the first algebraic construction of hierarchical quasi-cyclic codes from Reed-Solomon codes, with detailed parameters and bounds, advancing the theoretical understanding of these codes.

Findings

Codes with Tanner graph girth 6 from k=2 Reed-Solomon codes

Some codes meet the best known minimum distance for binary codes

New algebraic bounds on code parameters

Abstract

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.