Moderate Density Parity-Check Codes from Projective Bundles

TL;DR

This paper introduces a new class of moderate density parity-check codes constructed from finite geometry, specifically projective bundles, with proven minimum distance, dimension, and optimized error-correction performance.

Contribution

It presents a novel construction of MDPC codes using projective geometry and analyzes their properties and decoding performance, which was not previously explored.

Findings

Codes have a natural quasi-cyclic structure.

Minimum distance and dimension are explicitly determined.

Achieve optimal error-correction performance in one decoding round.

Abstract

A new construction for moderate density parity-check (MDPC) codes using finite geometry is proposed. We design a parity-check matrix for this family of binary codes as the concatenation of two matrices: the incidence matrix between points and lines of the Desarguesian projective plane and the incidence matrix between points and ovals of a projective bundle. A projective bundle is a special collection of ovals which pairwise meet in a unique point. We determine minimum distance and dimension of these codes, showing that they have a natural quasi-cyclic structure. In addition, we analyze the error-correction performance within one round of a modification of Gallager's bit-flipping decoding algorithm. In this setting, our codes have the best possible error-correction performance for this range of parameters.