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Pseudoredundancy for the Bit-Flipping Algorithm

Jens Zumbrägel

TL;DR

This work introduces bit-flipping pseudoredundancy, defined as the minimum number of parity-check rows required for the bit-flipping algorithm to correct up to $\frac{d-1}{2}$ errors in a binary code of minimum distance $d$, and investigates its properties in codes derived from finite geometries. It combines expander-graph insights, incidence-geometry constructions, and spectral bounds to derive concrete pseudoredundancy and decoding guarantees for Hamming, simplex, projective-plane, and partial-geometry codes, including explicit bounds such as $\rho \le n$ in several cases and $w \ge q+2$ via eigenvalue analysis. The results connect combinatorial design structure to decoding performance, offering practical guidance for finite-length decoding in settings like post-quantum cryptography (e.g., BIKE) and advancing the understanding of redundancy notions related to stopping sets and pseudocodewords. Open questions remain about codes with infinite pseudoredundancy and extensions to enhanced bit-flipping strategies.

Abstract

The analysis of the decoding failure rate of the bit-flipping algorithm has received increasing attention. For a binary linear code we consider the minimum number of rows in a parity-check matrix such that the bit-flipping algorithm is able to correct errors up to the minimum distance without any decoding failures. We initiate a study of this bit-flipping redundancy, which is akin to the stopping set, trapping set or pseudocodeword redundancy of binary linear codes, and focus in particular on codes based on finite geometries.

Pseudoredundancy for the Bit-Flipping Algorithm

TL;DR

This work introduces bit-flipping pseudoredundancy, defined as the minimum number of parity-check rows required for the bit-flipping algorithm to correct up to errors in a binary code of minimum distance , and investigates its properties in codes derived from finite geometries. It combines expander-graph insights, incidence-geometry constructions, and spectral bounds to derive concrete pseudoredundancy and decoding guarantees for Hamming, simplex, projective-plane, and partial-geometry codes, including explicit bounds such as in several cases and via eigenvalue analysis. The results connect combinatorial design structure to decoding performance, offering practical guidance for finite-length decoding in settings like post-quantum cryptography (e.g., BIKE) and advancing the understanding of redundancy notions related to stopping sets and pseudocodewords. Open questions remain about codes with infinite pseudoredundancy and extensions to enhanced bit-flipping strategies.

Abstract

The analysis of the decoding failure rate of the bit-flipping algorithm has received increasing attention. For a binary linear code we consider the minimum number of rows in a parity-check matrix such that the bit-flipping algorithm is able to correct errors up to the minimum distance without any decoding failures. We initiate a study of this bit-flipping redundancy, which is akin to the stopping set, trapping set or pseudocodeword redundancy of binary linear codes, and focus in particular on codes based on finite geometries.
Paper Structure (12 sections, 7 theorems, 6 equations, 1 figure)

This paper contains 12 sections, 7 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parity-Check Codes
  4. Bit-Flipping Algorithm
  5. Expander Codes
  6. Codes Based on Finite Geometries
  7. The Hamming and Simplex Codes
  8. Eigenvalue Analysis
  9. Decoding Failure
  10. Two Errors
  11. Three Errors
  12. More Errors

Key Result

Theorem 1

A $(c, d, \frac{c}{2})$-expander code has minimum distance greater than $d$.

Figures (1)

  • Figure 1: Example run of the bit-flipping algorithm. The white and gray disks are the satisfied and unsatisfied parity-checks, respectively. The squares represent a block of parity-checks that is being affected by a bit-flip, while the given number of total unsatisfied parity-checks is decreasing.

Theorems & Definitions (17)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Example 1
  • Proposition 1
  • proof
  • Example 2
  • Remark 1
  • Proposition 2
  • proof
  • ...and 7 more