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Dependence Analysis and Structured Construction for Batched Sparse Code

Jiaxin Qing, Xiaohong Cai, Yijun Fan, Mingyang Zhu, Raymond W. Yeung

TL;DR

A hardware-friendly structured BATS code called the Cyclic-Shift BATS (CS-BATS) code is proposed, which constructs the code from a small base graph using light-weight cyclic-shift operations and demonstrates that when the base graph is properly designed, a higher decoding rate and a smaller complexity can be achieved compared with the random BATS code.

Abstract

In coding theory, codes are usually designed with a certain level of randomness to facilitate analysis and accommodate different channel conditions. However, the resulting random code constructed can be suboptimal in practical implementations. Represented by a bipartite graph, the Batched Sparse Code (BATS Code) is a randomly constructed erasure code that utilizes network coding to achieve near-optimal performance in wireless multi-hop networks. In the performance analysis in the previous research, it is implicitly assumed that the coded batches in the BATS code are independent. This assumption holds only asymptotically when the number of input symbols is infinite, but it does not generally hold in a practical setting where the number of input symbols is finite, especially when the code is constructed randomly. We show that dependence among the batches significantly degrades the code's performance. In order to control the batch dependence through graphical design, we propose constructing the BATS code in a structured manner. A hardware-friendly structured BATS code called the Cyclic-Shift BATS (CS-BATS) code is proposed, which constructs the code from a small base graph using light-weight cyclic-shift operations. We demonstrate that when the base graph is properly designed, a higher decoding rate and a smaller complexity can be achieved compared with the random BATS code.

Dependence Analysis and Structured Construction for Batched Sparse Code

TL;DR

A hardware-friendly structured BATS code called the Cyclic-Shift BATS (CS-BATS) code is proposed, which constructs the code from a small base graph using light-weight cyclic-shift operations and demonstrates that when the base graph is properly designed, a higher decoding rate and a smaller complexity can be achieved compared with the random BATS code.

Abstract

In coding theory, codes are usually designed with a certain level of randomness to facilitate analysis and accommodate different channel conditions. However, the resulting random code constructed can be suboptimal in practical implementations. Represented by a bipartite graph, the Batched Sparse Code (BATS Code) is a randomly constructed erasure code that utilizes network coding to achieve near-optimal performance in wireless multi-hop networks. In the performance analysis in the previous research, it is implicitly assumed that the coded batches in the BATS code are independent. This assumption holds only asymptotically when the number of input symbols is infinite, but it does not generally hold in a practical setting where the number of input symbols is finite, especially when the code is constructed randomly. We show that dependence among the batches significantly degrades the code's performance. In order to control the batch dependence through graphical design, we propose constructing the BATS code in a structured manner. A hardware-friendly structured BATS code called the Cyclic-Shift BATS (CS-BATS) code is proposed, which constructs the code from a small base graph using light-weight cyclic-shift operations. We demonstrate that when the base graph is properly designed, a higher decoding rate and a smaller complexity can be achieved compared with the random BATS code.
Paper Structure (20 sections, 2 theorems, 22 equations, 8 figures, 2 tables, 3 algorithms)

This paper contains 20 sections, 2 theorems, 22 equations, 8 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. BATS Code Preliminary
  4. Encoding
  5. Recoding
  6. Decoding
  7. Degree Distribution Optimization
  8. Decoding Rate and Dependence
  9. Correlation Decreases Decodable Expectation
  10. Dependence Bounds Decodable Probability
  11. Implication of Asymptotic Assumption in Dependence
  12. Structured BATS Code
  13. Construction of Batches
  14. Bounded Complexity
  15. Preserved Properties
  16. ...and 5 more sections

Key Result

Lemma 1

Assume that a variable node is connected to $n$ check nodes, $\tilde{C}_{1}, \tilde{C}_{2}, ..., \tilde{C}_{n}$, where $n\geq1$. Then for all $i\in [1,n]$ and all $\mathcal{J}\subseteq[1,n]\backslash\{i\}$,

Figures (8)

  • Figure 1: Graphical Representation of the BATS Code. The input symbols are represented by circles (variable nodes). The batches are represented by squares (check nodes), which consist of several coded symbols.
  • Figure 2: Random code construction complicates circuit routing in hardware implementations (A high-level illustration). Comparision of the circuits between (a) randomly selecting input symbols for encoding and (b) selecting input symbols in a structured way using shift operations. Random selection leads to a fully connected circuit implementation in hardware.
  • Figure 3: Four cases of Pearson correlation ($\rho$) for ($C_1, C_2$) in multiple trials, which fully characterize the relationship between two Bernoulli random variables. The darkness of the color indicates occurrence frequency. A negative correlation is physically invalid in the setting of a BATS code.
  • Figure 4: A variable node (VN) is connected to multiple check nodes (CNs). All other connections are arbitrary.
  • Figure 5: Tree representation of a BATS code. The root variable node, V0, is connected to two check nodes, C1 and C2, through connections p1 and p2. Only 4 levels are shown. Belief Propagation decoding is performed level by level from the leaves to the root.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Theorem 1
  • Definition 1
  • Definition 2