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Moderate-density parity-check codes from projective bundles.

Jessica Bariffi1,2, Sam Mattheus3, Alessandro Neri4

  • 1Institute of Mathematics, University of Zurich, Zurich, Switzerland.

Designs, Codes, and Cryptography
|November 18, 2022
PubMed
Summary
This summary is machine-generated.

New moderate-density parity-check (MDPC) codes are constructed using finite geometry. These novel binary codes exhibit optimal error-correction performance with a modified bit-flipping decoding algorithm.

Keywords:
Bit-flipping decoding algorithmMDPC codesProjective bundleProjective plane

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Area of Science:

  • Coding Theory
  • Finite Geometry
  • Information Theory

Background:

  • Moderate-density parity-check (MDPC) codes are crucial for efficient data transmission.
  • Finite geometry offers a powerful framework for constructing error-correcting codes.
  • Existing MDPC code constructions require further optimization for performance.

Purpose of the Study:

  • To propose novel constructions for moderate-density parity-check (MDPC) codes.
  • To leverage finite geometry, specifically Desarguesian projective planes and projective bundles, for code design.
  • To analyze the properties and error-correction capabilities of the newly constructed codes.

Main Methods:

  • Designing parity-check matrices by concatenating incidence matrices from finite geometry.
  • Utilizing incidence matrices of Desarguesian projective planes and projective bundles.
  • Analyzing code properties such as minimum distance and dimension.
  • Employing a modified Gallager's bit-flipping decoding algorithm for performance evaluation.

Main Results:

  • Successfully constructed new families of binary MDPC codes.
  • Determined the minimum distance and dimension for these codes.
  • Demonstrated a natural quasi-cyclic structure within the proposed codes.
  • Showcased superior error-correction performance after one decoding round compared to existing methods.

Conclusions:

  • The proposed finite geometry-based MDPC codes offer a promising new direction in coding theory.
  • These codes achieve excellent error-correction performance, particularly with bit-flipping decoding.
  • The quasi-cyclic structure simplifies decoding and implementation.