A class of quasi-cyclic binary parity-check codes from Reed-Solomon codes

In this paper, we explore a construction of binary parity-check codes from nonbinary codes, with a specific focus on Reed-Solomon codes. The parity-check matrices of these codes come from the 1964 construction of superimposed codes by Kautz and Singleton. While some basic bounds on parameters of these parity-check codes are known, the specific parameters are highly dependent on the nonbinary code that one starts with. In this paper, we delve more deeply into parameters of this new class of girth six codes, showing that they are quasi-cyclic, providing bounds on the dimensions of the codes, and bounds on minimum distances in some specific cases. We present a table of small code parameters and note that some of these codes meet the best known minimum distance for binary codes. 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.