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On Streaming Codes for Simultaneously Correcting Burst and Random Erasures

Shobhit Bhatnagar, Biswadip Chakraborty, P. Vijay Kumar

TL;DR

This paper addresses streaming codes over sliding-window channels that admit either a burst erasure of length $\le b$ with $\le e$ random erasures or $\le a$ random erasures within each window, under a delay constraint $\tau$. It analyzes DE-based constructions and derives the optimal rate $R_{\text{opt}}=\frac{w-(b+e)}{w}$, achievable via DE of a systematic $[w,w-(b+e)]$ code, with an $O(w)$ field size, and shows under the MDS conjecture that the field-size cannot be significantly reduced when $e>1$. For the special case $e=1$, it provides a sub-linear field-size construction (Construction I) that yields a family of $(b_1,b_2)$-codes meeting the rate bound in certain regimes, with field-size optimality or near-optimality under sparsity constraints. Additionally, it derives an upper bound on the minimum distance of cyclic codes, and characterizes cyclic codes that attain this bound via their burst-and-random erasure recovery properties. These results map precise rate-field-size trade-offs and illuminate the role of cyclic structure in joint burst/random erasure correction for streaming communications.

Abstract

Streaming codes are packet-level codes that recover dropped packets within a strict decoding-delay constraint. We study streaming codes over a sliding-window (SW) channel model which admits only those erasure patterns which allow either a single burst erasure of $\le b$ packets along with $\le e$ random packet erasures, or else, $\le a$ random packet erasures, in any sliding-window of $w$ time slots. We determine the optimal rate of a streaming code constructed via the popular diagonal embedding (DE) technique over such a SW channel under delay constraint $τ=(w-1)$ and provide an $O(w)$ field size code construction. For the case $e>1$, we show that it is not possible to significantly reduce this field size requirement, assuming the well-known MDS conjecture. We then provide a block code construction whose DE yields a streaming code achieving the rate derived above, over a field of size sub-linear in $w,$ for a family of parameters having $e=1.$ We show the field size optimality of this construction for some parameters, and near-optimality for others under a sparsity constraint. Additionally, we derive an upper-bound on the $d_{\text{min}}$ of a cyclic code and characterize cyclic codes which achieve this bound via their ability to simultaneously recover from burst and random erasures.

On Streaming Codes for Simultaneously Correcting Burst and Random Erasures

TL;DR

This paper addresses streaming codes over sliding-window channels that admit either a burst erasure of length with random erasures or random erasures within each window, under a delay constraint . It analyzes DE-based constructions and derives the optimal rate , achievable via DE of a systematic code, with an field size, and shows under the MDS conjecture that the field-size cannot be significantly reduced when . For the special case , it provides a sub-linear field-size construction (Construction I) that yields a family of -codes meeting the rate bound in certain regimes, with field-size optimality or near-optimality under sparsity constraints. Additionally, it derives an upper bound on the minimum distance of cyclic codes, and characterizes cyclic codes that attain this bound via their burst-and-random erasure recovery properties. These results map precise rate-field-size trade-offs and illuminate the role of cyclic structure in joint burst/random erasure correction for streaming communications.

Abstract

Streaming codes are packet-level codes that recover dropped packets within a strict decoding-delay constraint. We study streaming codes over a sliding-window (SW) channel model which admits only those erasure patterns which allow either a single burst erasure of packets along with random packet erasures, or else, random packet erasures, in any sliding-window of time slots. We determine the optimal rate of a streaming code constructed via the popular diagonal embedding (DE) technique over such a SW channel under delay constraint and provide an field size code construction. For the case , we show that it is not possible to significantly reduce this field size requirement, assuming the well-known MDS conjecture. We then provide a block code construction whose DE yields a streaming code achieving the rate derived above, over a field of size sub-linear in for a family of parameters having We show the field size optimality of this construction for some parameters, and near-optimality for others under a sparsity constraint. Additionally, we derive an upper-bound on the of a cyclic code and characterize cyclic codes which achieve this bound via their ability to simultaneously recover from burst and random erasures.
Paper Structure (11 sections, 1 theorem, 7 equations, 3 figures)

This paper contains 11 sections, 1 theorem, 7 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. Notation
  4. Streaming Codes
  5. Framework
  6. Diagonal Embedding
  7. Preliminaries
  8. Optimal Rate of $(a,(b,e),w,\tau=w-1)$ Streaming Codes Constructed via DE
  9. A Sub-Linear Field Size Construction for A class of Parameters where $e=1$
  10. An Upper-Bound on the Minimum Distance of Cyclic Codes
  11. Proof of Theorem \ref{['thm:sub-linear_field_size']}

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 $E\subseteq[0:n-1]$ be erased from $\mathcal{C}$. Then, $\mathcal{C}$ can recover all the erased code symbols iff $\{H_i\mid i\in E\}$ is a linearly independent set.

Figures (3)

  • Figure 1: An erasure pattern that is admissible in the $(2,(3,1),5)$-SW channel. Colored squares denote erasures.
  • Figure 2: DE of a $[6,3]$ 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: A periodic erasure pattern admissible in the $(a,(b,e),w)$-SW channel when $e\ge (b-1)$. Colored squares denote erasures.

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • proof
  • proof
  • Remark 1
  • proof
  • proof
  • proof
  • proof