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Non-Binary Covering Codes for Low-Access Computations

Vinayak Ramkumar, Netanel Raviv, Itzhak Tamo

TL;DR

This work explores non-binary covering codes and develop schemes that outperform the state-of-the-art for some coefficient sets and provides a more general coefficient complexity definition and shows its applicability to the access- redundancy tradeoff.

Abstract

Given a real dataset and a computation family, we wish to encode and store the dataset in a distributed system so that any computation from the family can be performed by accessing a small number of nodes. In this work, we focus on the families of linear computations where the coefficients are restricted to a finite set of real values. For two-valued computations, a recent work presented a scheme that gives good feasible points on the access-redundancy tradeoff. This scheme is based on binary covering codes having a certain closure property. In a follow-up work, this scheme was extended to all finite coefficient sets, using a new additive-combinatorics notion called coefficient complexity. In the present paper, we explore non-binary covering codes and develop schemes that outperform the state-of-the-art for some coefficient sets. We provide a more general coefficient complexity definition and show its applicability to the access-redundancy tradeoff.

Non-Binary Covering Codes for Low-Access Computations

TL;DR

This work explores non-binary covering codes and develop schemes that outperform the state-of-the-art for some coefficient sets and provides a more general coefficient complexity definition and shows its applicability to the access- redundancy tradeoff.

Abstract

Given a real dataset and a computation family, we wish to encode and store the dataset in a distributed system so that any computation from the family can be performed by accessing a small number of nodes. In this work, we focus on the families of linear computations where the coefficients are restricted to a finite set of real values. For two-valued computations, a recent work presented a scheme that gives good feasible points on the access-redundancy tradeoff. This scheme is based on binary covering codes having a certain closure property. In a follow-up work, this scheme was extended to all finite coefficient sets, using a new additive-combinatorics notion called coefficient complexity. In the present paper, we explore non-binary covering codes and develop schemes that outperform the state-of-the-art for some coefficient sets. We provide a more general coefficient complexity definition and show its applicability to the access-redundancy tradeoff.
Paper Structure (12 sections, 7 theorems, 25 equations, 1 figure)

This paper contains 12 sections, 7 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Problem Statement
  4. Covering Codes
  5. Prior Results
  6. Schemes based on Non-Binary Codes
  7. Three-valued Computations
  8. Entire space
  9. Repetition code
  10. Hamming code
  11. Expanded Hamming code and its amalgamations
  12. Comparison with known results

Key Result

Theorem 1

If there exists a (not necessarily linear) code $\mathcal{C}$ of length $m$ and covering radius $r$ over an alphabet $\Sigma$, then the pair $(\frac{m+|\mathcal{C}|}{m},\frac{r+1}{m})$ is $\mathcal{A}$-feasible, where $\mathcal{A} \subset \mathbb{R}$ is any set of size $|\Sigma|$.

Figures (1)

  • Figure 1: The plot depicts access-redundancy tradeoff for $\mathcal{A}=\{0, \pm1\}$. The points in blue are the new solutions obtained using ternary covering codes, whereas red is used to indicate the best-known systematic solutions in the literature RamkumarRT23_AllertonRamkumarRT23_ISIT.

Theorems & Definitions (17)

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