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On Streaming Codes for Burst and Random Errors

Shobhit Bhatnagar, P. Vijay Kumar

TL;DR

The paper investigates streaming codes that correct burst and random errors under strict decoding delays. It establishes a key equivalence between SC_ERRs and erasure-based SCs, deriving the optimal rate R_opt = (w−2a)/w for random errors and providing a rate-optimal DE construction from [w,w−2a] for τ = w−1. For multiple bursts, it proves a necessary divisibility constraint and presents a tight rate upper bound that can be achieved by diagonal embedding when the divisibility condition holds; it also links SC_ERRs for burst errors to SCs with doubled burst capacity. Collectively, these results clarify the fundamental limits and constructive approaches for rate-optimal, low-latency streaming codes under burst and random error models and show when DE-based designs are sufficient.

Abstract

Streaming codes (SCs) are packet-level codes that recover erased packets within a strict decoding-delay deadline. Streaming codes for various packet erasure channel models such as sliding-window (SW) channel models that admit random or burst erasures in any SW of a fixed length have been studied in the literature, and the optimal rate as well as rate-optimal code constructions of SCs over such channel models are known. In this paper, we study error-correcting streaming codes ($\text{SC}_{\text{ERR}}$s), i.e., packet-level codes which recover erroneous packets within a delay constraint. We study $\text{SC}_{\text{ERR}}$s for two classes of SW channel models, one that admits random packet errors, and another that admits multiple bursts of packet errors, in any SW of a fixed length. For the case of random packet errors, we establish the equivalence of an $\text{SC}_{\text{ERR}}$ and a corresponding SC that recovers from random packet erasures, thus determining the optimal rate of an $\text{SC}_{\text{ERR}}$ for this setting, and providing a rate-optimal code construction for all parameters. We then focus on SCs that recover from multiple erasure bursts and derive a rate-upper-bound for such SCs. We show the necessity of a divisibility constraint for the existence of an SC constructed by the popular diagonal embedding technique, that achieves this rate-bound under a stringent delay requirement. We then show that a construction known in the literature achieves this rate-bound when the divisibility constraint is met. We further show the equivalence of the SCs considered and $\text{SC}_{\text{ERR}}$s for the setting of multiple error bursts, under a stringent delay requirement.

On Streaming Codes for Burst and Random Errors

TL;DR

The paper investigates streaming codes that correct burst and random errors under strict decoding delays. It establishes a key equivalence between SC_ERRs and erasure-based SCs, deriving the optimal rate R_opt = (w−2a)/w for random errors and providing a rate-optimal DE construction from [w,w−2a] for τ = w−1. For multiple bursts, it proves a necessary divisibility constraint and presents a tight rate upper bound that can be achieved by diagonal embedding when the divisibility condition holds; it also links SC_ERRs for burst errors to SCs with doubled burst capacity. Collectively, these results clarify the fundamental limits and constructive approaches for rate-optimal, low-latency streaming codes under burst and random error models and show when DE-based designs are sufficient.

Abstract

Streaming codes (SCs) are packet-level codes that recover erased packets within a strict decoding-delay deadline. Streaming codes for various packet erasure channel models such as sliding-window (SW) channel models that admit random or burst erasures in any SW of a fixed length have been studied in the literature, and the optimal rate as well as rate-optimal code constructions of SCs over such channel models are known. In this paper, we study error-correcting streaming codes (s), i.e., packet-level codes which recover erroneous packets within a delay constraint. We study s for two classes of SW channel models, one that admits random packet errors, and another that admits multiple bursts of packet errors, in any SW of a fixed length. For the case of random packet errors, we establish the equivalence of an and a corresponding SC that recovers from random packet erasures, thus determining the optimal rate of an for this setting, and providing a rate-optimal code construction for all parameters. We then focus on SCs that recover from multiple erasure bursts and derive a rate-upper-bound for such SCs. We show the necessity of a divisibility constraint for the existence of an SC constructed by the popular diagonal embedding technique, that achieves this rate-bound under a stringent delay requirement. We then show that a construction known in the literature achieves this rate-bound when the divisibility constraint is met. We further show the equivalence of the SCs considered and s for the setting of multiple error bursts, under a stringent delay requirement.
Paper Structure (22 sections, 5 theorems, 9 equations, 4 figures)

This paper contains 22 sections, 5 theorems, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background and Related Prior Work
  3. Erasure-Correcting Streaming Codes
  4. Framework MartSunTIT04
  5. $(a,w)$-SW Channel
  6. Multiple Burst-Erasure Sliding Window Channel
  7. Diagonal Embedding
  8. Error-Correcting Streaming Codes
  9. Framework
  10. $(a,w)$-Error SW Channel
  11. Multiple Burst-Error Sliding Window Channel
  12. Preliminaries
  13. Optimal Rate of $(a,w,\tau)$-$\text{SC}_{\text{\tiny ERR}}$s
  14. Streaming Codes for Multiple Erasure and Error Bursts
  15. Streaming Codes for Multiple Erasure Bursts
  16. ...and 7 more sections

Key Result

Lemma 1

Let $\mathcal{C}$ be an $[n,k]$ code and let $H$ be an $(n-k)\times n$ parity-check matrix of $\mathcal{C}$. Let the coordinates in $\mathcal{E}\subseteq[0:n-1]$ be erased from $\mathcal{C}$, and let $i\in\mathcal{E}$. Then, the $i$-th code symbol in any codeword can be recovered from the code symbo

Figures (4)

  • Figure 1: An admissible erasure pattern of the $(3,2,7)$-MBSW channel. Colored squares denote erasures.
  • Figure 2: DE of a $[5, 2]$ systematic code $\mathcal{C}$. Each column is a coded packet and every diagonal of the same color is a codeword of $\mathcal{C}$.
  • Figure 3: Setting of error-correcting streaming codes.
  • Figure 4: An admissible periodic erasure pattern of the $(z,b,w)$-MBSW channel.

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • proof
  • proof
  • Remark 1