LNCS 2764 Bounds on 2 Query Codeword Testing 1st Edition by Eli BenSasson, Oded Goldreich, Madhu Sudan ISBN 3540380450 9783540380450

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Product details:

ISBN 10: 3540380450

ISBN 13: 9783540380450

Author:  Eli Ben-Sasson, Oded Goldreich, Madhu Sudan 

LNCS 2764 – Bounds on 2-Query Codeword Testing 1st Edition: We present upper bounds on the size of codes that are locally testable by querying only two input symbols. For linear codes, we show that any 2-locally testable code with minimal distance δn over any finite field <span id="MathJax-Element-1-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="F”>F cannot have more than <span id="MathJax-Element-2-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="|F|3/δ”>|F|3/δ codewords. This result holds even for testers with two-sided error. For general (non-linear) codes we obtain the exact same bounds on the code size as a function of the minimal distance, but our bounds apply only for binary alphabets and one-sided error testers (i.e. with perfect completeness). Our bounds are obtained by examining the graph induced by the set of possible pairs of queries made by a codeword tester on a given code. We also demonstrate the tightness of our upper bounds and the essential role of certain parameters.

 

LNCS 2764 – Bounds on 2-Query Codeword Testing 1st Edition Table of contents:

1 Introduction

2 Formal Setting

2.1 Codes

2.2 Testers and Tests

2.3 Our Results

3 Linear Codes

3.1 Linearity Tests and Dual Codes

3.2 Upper Bounds on Code Size

3.3 Tightness of the Upper Bound

4 Non-linear Codes

 

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