GRAPH CODES AND THEIR APPLICATIONS 2014 DRAFT Incremental Redundancy, Fountain Codes and Advanced Topics Suayb S. Arslan 141 Innovation Dr. Irvine, CA 92617
[email protected] “All men by nature desire knowledge." Aristotle Version 0.2 – July 2014 Revision History: • Jan. 2014, First version is released. • Feb. 2014, Uploaded on arXiv.org for public access. • Jul. 2014, Version 0.2 is released. – Table of Contents is added for easy reference. – Fixes few grammatical errors, changes the organization and extends the section of systematic fountain code constructions arXiv:1402.6016v2 [cs.IT] 14 Jul 2014 1 GRAPH CODES AND THEIR APPLICATIONS 2014 DRAFT CONTENTS 1 Abstract 3 2 Introduction 4 2.1 Notation ........................................ 4 2.2 ChannelModelandLinearCodes . ..... 5 2.3 IncrementalRedundancy . ..... 6 3 Linear Fountain Codes 8 3.1 High density Fountain codes: Maximum Likelihood Decoding .......... 11 3.2 Low Density Fountain codes: BP decoding and LT Codes . .......... 14 3.3 Asymptotical performance of LT codes under BP Decoding . ........... 20 3.4 Expected Ripple size and Degree Distribution Design . ............. 23 3.5 Systematic Constructions of LT codes . ......... 26 3.6 Generalizations/Extensions of LT codes: Unequal Error Protection . 27 4 Concatenated Linear Fountain Codes 29 4.1 Asymptotically good degree distributions . ............ 30 4.2 Precoding ....................................... 32 4.2.1 TornadoCodesasPrecodes . 33 4.2.2 LDPCCodesasPrecodes . 34 4.2.3 HammingCodesasPrecodes . 36 4.2.4 Repeat and Accumulate Codes as Precodes . ...... 38 4.3 Concatenated Fountain Code Failure Probability . ............ 38 4.4 Systematic Constructions of Concantenated Fountain Codes ........... 40 5 Advanced Topics 45 5.1 Connectionstotherandomgraphtheory . ........ 46 5.2 InactivationDecoding . ...... 49 5.3 Standardized Fountain Codes for Data Streaming .