Abstract

Dyadic matrices are a class of reproducible matrices whose entries are completely determined by a single row, while quasi-dyadic matrices extend this structure through block arrays of dyadic matrices. Although related structured code families, such as quasi-cyclic codes, have been studied extensively, comparatively less is known about the Tanner graph structure of codes constructed from dyadic and quasi-dyadic matrices.

In this work, we study how the algebraic structure of dyadic and quasi-dyadic matrices determines the local structure of their Tanner graphs, with the goal of using these properties for code design. We characterize graph isomorphisms induced by the dyadic structure and derive conditions governing the occurrence of short cycles and absorbing sets. We then incorporate these structural conditions into a progressive edge-growth (PEG) construction tailored to quasi-dyadic codes, allowing undesirable local graph configurations to be avoided during code construction. Finally, we compare the resulting code constructions under belief propagation decoding and examine the relationship between their Tanner graph structure and decoding performance. This is joint work with Anthony Gómez-Fonseca, Gretchen Matthews, and Tefjol Pllaha.