Codes based on dyadic matrices and their generalizations

Codes based on algebraically structured parity check matrices are desirable in applications due to their compact representations, practical implementation advantages, and tractable decoder performance analysis. In this paper, we examine codes based on parity check matrices that are dyadic, $ n $-adic, or quasi-dyadic (QD), meaning the parity check matrix representation is block structured with dyadic matrices as blocks. Depending on the number of nonzero positions in the leading row of each matrix or block, these codes may be either low density or moderate density. We examine basic code properties of dyadic, $ n $-adic, and QD parity check codes, including reproducibility, bounds on the dimension and minimum distance, cycle structure of the corresponding Tanner graph, and their possible use in quantum code constructions.