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Majority-Logic Decoding of Binary Locally Recoverable Codes: A Probabilistic Analysis

Hoang Ly, Emina Soljanin, Philip Whiting

TL;DR

The paper investigates the probabilistic decoding performance of binary linear locally recoverable codes with fixed locality $r$ and growing availability $t$ under majority-logic decoding over the Binary Symmetric Channel ($BSC$) and Binary Erasure Channel ($BEC$). By deriving explicit bounds on decoding failure probabilities, BER, and BLER as functions of $(r,t)$ and channel parameters, it shows that, for $t(n)=\omega(\log n)$, the block error rate vanishes asymptotically and most error/erasure patterns of weight linear in $n$ are correctable. The analysis contrasts worst-case (deterministic) guarantees with typical (probabilistic) performance, highlighting a substantial gap between adversarial limits and stochastic behavior. The results motivate practical use of majority-logic decoding with LRCs in latency-critical settings and point to future work on extending the probabilistic framework to $q$-ary LRCs.

Abstract

Locally repairable codes (LRCs) were originally introduced to enable efficient recovery from erasures in distributed storage systems by accessing only a small number of other symbols. While their structural properties-such as bounds and constructions-have been extensively studied, the performance of LRCs under random erasures and errors has remained largely unexplored. In this work, we study the error- and erasure-correction performance of binary linear LRCs under majority-logic decoding (MLD). Focusing on LRCs with fixed locality and varying availability, we derive explicit upper bounds on the probability of decoding failure over the memoryless Binary Erasure Channel (BEC) and Binary Symmetric Channel (BSC). Our analysis characterizes the behavior of the bit-error rate (BER) and block-error rate (BLER) as functions of the locality and availability parameters. We show that, under mild growth conditions on the availability, the block decoding failure probability vanishes asymptotically, and that majority-logic decoding can successfully correct virtually all of error and erasure patterns of weight linear in the blocklength. The results reveal a substantial gap between worst-case guarantees and typical performance under stochastic channel models.

Majority-Logic Decoding of Binary Locally Recoverable Codes: A Probabilistic Analysis

TL;DR

The paper investigates the probabilistic decoding performance of binary linear locally recoverable codes with fixed locality and growing availability under majority-logic decoding over the Binary Symmetric Channel () and Binary Erasure Channel (). By deriving explicit bounds on decoding failure probabilities, BER, and BLER as functions of and channel parameters, it shows that, for , the block error rate vanishes asymptotically and most error/erasure patterns of weight linear in are correctable. The analysis contrasts worst-case (deterministic) guarantees with typical (probabilistic) performance, highlighting a substantial gap between adversarial limits and stochastic behavior. The results motivate practical use of majority-logic decoding with LRCs in latency-critical settings and point to future work on extending the probabilistic framework to -ary LRCs.

Abstract

Locally repairable codes (LRCs) were originally introduced to enable efficient recovery from erasures in distributed storage systems by accessing only a small number of other symbols. While their structural properties-such as bounds and constructions-have been extensively studied, the performance of LRCs under random erasures and errors has remained largely unexplored. In this work, we study the error- and erasure-correction performance of binary linear LRCs under majority-logic decoding (MLD). Focusing on LRCs with fixed locality and varying availability, we derive explicit upper bounds on the probability of decoding failure over the memoryless Binary Erasure Channel (BEC) and Binary Symmetric Channel (BSC). Our analysis characterizes the behavior of the bit-error rate (BER) and block-error rate (BLER) as functions of the locality and availability parameters. We show that, under mild growth conditions on the availability, the block decoding failure probability vanishes asymptotically, and that majority-logic decoding can successfully correct virtually all of error and erasure patterns of weight linear in the blocklength. The results reveal a substantial gap between worst-case guarantees and typical performance under stochastic channel models.
Paper Structure (14 sections, 7 theorems, 59 equations, 2 figures)

This paper contains 14 sections, 7 theorems, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Locally Recoverable Codes with Availability
  3. Locally Recoverable Codes with Availability
  4. Majority-Logic Decoding
  5. Analysis of Decoding Capability of LRCs
  6. Deterministic versus Probabilistic Perspectives
  7. Deterministic approach.
  8. Probabilistic approach.
  9. Binary Symmetric Channel ($\mathrm{BSC}$)
  10. Binary Erasure Channel ($\mathrm{BEC}$)
  11. Simulations
  12. Bit Decoding Failure Probability
  13. Block Decoding Failure Probability
  14. Conclusion

Key Result

Theorem 1

Consider transmission over a $\mathrm{BSC}(p_f)$ with $\mathrm{i.i.d.}$ errors, where $0 \le p_f < 0.5$ is a fixed bit-flip probability. For binary linear $(r, t)_a$-LRC, the symbol (or bit) decoding failure probability of MLD satisfies

Figures (2)

  • Figure 1: Bit Decoding Failure Probability: Empirical vs Theoretical ($p_f=0.2, r=4$). The plot compares linear, polylogarithmic, and sub-logarithmic availability regimes.
  • Figure 2: Upper (Union) bound for Block Decoding Failure ($p_f=0.13, r=4$). The plot demonstrates the impact of the logarithmic exponent $\alpha$ on block decoding success.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Example 1
  • Theorem 1
  • proof
  • Lemma 1: Chernoff bound
  • proof
  • Remark 1: Exponential Tightness
  • Remark 2
  • Theorem 2: MLD Success for LRCs on the BSC
  • ...and 12 more