Parity-check Codes from Disjunct Matrices
Kathryn Haymaker, Emily McMillon
TL;DR
This work bridges combinatorial matrix properties and coding theory by showing that $D$-disjunct and $\bar{D}$-separable matrices yield parity-check codes with guaranteed minimum and stopping distances ($d_{min}, s_{min} \ge D+2$). It analyzes three algebraic constructions—Macula's subset construction, $t$-packing designs, and Kautz–Singleton expansions from $q$-ary codes—demonstrating that their Tanner graphs can achieve girth $6$ ($a_{\\max}=1$), which enables near-optimal one-round bit-flipping performance under a modified Gallager decoding approach. The paper provides concrete parameter results for rate, distance, girth, and density, highlighting MDPC regimes and exact or tight bounds on $d_{min}$ in several cases (notably Macula’s construction). By linking disjunct/separable properties to decoding performance and code parameters, it opens a design space for structured MDPC-like parity-check codes with potential applications in post-quantum cryptography and beyond.
Abstract
The matrix representations of linear codes have been well-studied for use as disjunct matrices. However, no connection has previously been made between the properties of disjunct matrices and the parity-check codes obtained from them. This paper makes this connection for the first time. We provide some fundamental results on parity-check codes from general disjunct matrices (in particular, a minimum distance bound). We then consider three specific constructions of disjunct matrices and provide parameters of their corresponding parity-check codes including rate, distance, girth, and density. We show that, by choosing the correct parameters, the codes we construct have the best possible error-correction performance after one round of bit-flipping decoding with regard to a modified version of Gallager's bit-flipping decoding algorithm.