Abstract

We continue the study of robustly testable tensor codes and expand the class of base codes that can be used as a starting point for the construction of locally testable codes via robustly testable tensor products. In particular, we show that all unique-neighbor expander codes and all locally correctable codes, when tensored with any other good-distance code, are robustly testable and hence can be used to construct locally testable codes. Previous work by Dinur et al. (2006) required stronger expansion properties to obtain locally testable codes. Our proofs follow by defining the notion of weakly smooth codes that generalize the smooth codes of Dinur et al. We show that weakly smooth codes are sufficient for constructing robustly testable tensor codes. Using the weaker definition, we are able to expand the family of base codes to include the aforementioned ones.

Highlights

  • A linear code over a finite field F is a linear subspace C ⊆ Fn

  • One of the simplest ways to compose codes for the construction of Locally Testable Codes (LTCs) is by use of the tensor product, as suggested by Ben-Sasson and Sudan [2]

  • The minimal requirement in terms of expansion needed to argue an expander code has good (i. e., constant relative) distance, using currently known techniques, so our work shows all “combinatorially good” expander codes1 can be used for the construction of robustly testable tensor products

Read more

Summary

Introduction

A linear code over a finite field F is a linear subspace C ⊆ Fn. A code is locally testable if given a word x ∈ Fn one can verify whether x ∈ C by reading only a few (randomly chosen) symbols from x. One of the simplest ways to compose codes for the construction of LTCs is by use of the tensor product, as suggested by Ben-Sasson and Sudan [2] They introduced the notion of robust LTCs: An LTC is called robustly testable if whenever the received word is far from the code, the “view” of the tester is far from an accepting view with noticeable probability (see Definition 2.1). In comparison, [6] required stronger expansion parameters (γ < 1/4) of the kind needed to ensure an expander code is not merely good in terms of its distance, but can be decoded in linear time [15] Another family of codes shown here to result in robustly testable tensor products of pairs of codes is the family of locally correctable codes (LCCs), see Definition 7.1. The last two sections prove that unique-neighbor expander codes and locally correctable codes are weakly smooth

Preliminary Definitions
Tensor Product of Codes
Robust Locally Testable Codes
Main Results
Weakly Smooth codes
Smooth codes are also weakly smooth
Unique-Neighbor Expander Codes are weakly smooth
Locally correctable codes are weakly smooth
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call