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Tutorial on Reed-Solomon Error Correction Coding

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Tutorial on Reed-Solomon Error Correction Coding Lyndon B. Johnson Space Center Houston, Texas 77058 Technical Support Package Tutorial on Reed-Solomon Error Correction Coding NASA- Tech Briefs MSC-21834 NationalAeronautics and Space AdnTinistration Technical Support Package for TUTORIAL ON REED-SOLOMON ERROR CORRECTION COIDING MSC-21 834 , NASA Tech Briefs The information in this Technical Support Package comprises the docunnentation referenced in MSC-21 834 of NASA Tech Briefs. It is provided under the Technology Transfer Program of the National Aeronautics and Space Administration to make available the results of aerospace-related developments considered to have wider technological, scientific, or commercial applications. Further assistance is available from sources listed in NASA Tech Briefs on the page entitled “How You Can Benefit From NASA’s Technology Utilization Services.” Additional information regarding research and technology in this general area may be found in Scientific and Technical Aerospace Reports (STAR), which is a comprehensive abstracting and indexing journal covering worldwide repofl literature on the science and technology of space and aeronautics. STAR is available to the public on subscription from Registration Services NASA Center for AeroSpace Information 800 Elkridge Landing Road Linthicum Heights, MD 21090-2934 Telephone: (301) 621-0390, Fax: (301) 621-0134, E-mail: help @sti.nasa.gov NOTICE: This document was prepared under the sponsorship of the National Aeronautics and Space Administration. Neither the United States Government nor any person acting on behalf of the United States Government assumes any liability resulting from the use of the information contained in this document or warrants that such use will be free from privately owned rights. Chapter ABSTRACT . 1 INTRODUCTION . 2 1 GALOIS FIELDALGEBRA . 8 1.1 GROUPS . 8 1.2 FIELDS . 9 1.3 BINARY FIELD GF(2) . .................10 1.3.1 Bina~ Group . ...’ . ...10 1.3.2 Binary Field . 12 1.4 Extension FlELDsGF(2m) . ...14 1.4.1 ‘ Primitive Polvnomialsp(x]. ...14 1.4.2 - Field Symbols (xi . ...17 1.4.3 Different Svmbol Representations. ...21 1.4.4 isomorphic GF(2m)lmplementations . ...22 1.4.5 Addition and Subtraction Within GF(2m ) . .26 1.4.6 Multiplication and Division Within GF(2m) . 28 1.5 DIFFERENTALGEBRAICSTRUCTURES . ...30 1.6 SUMMARY . ...31 2 BLOCK CODES . ...32 2,1 BLOCKERRORCORRECTIONCODINGSYSTEM . ...3 2 2.2 APERFECT (3,1) BLOCKCODE. ...33 2.3 LINEARBLOCKCODES . ...39 2.4 SOMEMORE REL4TEDTOPICS . ...43 2.5 WHITE, BIACK, ANDGRAYZONES . ...46 2.6 SUMMARY . ...47 3 REED-SOLOMONENCODING . ...48 3.1 REED-SOLOMONENCODER . .-, . ...48 3.2 (n,k)RS CODES . ...........................5o 3.3 (15,9)RS PARAMETERS ..........................................51 3.3.1 Generator Polynomial q(x).... ...53 3.3.2 Code Word Polynomial C(X).., . ...54 3.4 SUMMARY . .,.....56 4 REED-SOLOMONDECODING . .,.....58 4.1 REED-SOLOMON DECODER . ...59 4.2 SYNDROMES . ...61 4.2.1 M@thodl :Svndrome Compone~ Si b. ,...........62 —— --i-- .\Isc’-2ls34 4,2.2 Method 2: Syndrome Polynomials(X) . ..63 4.3 ERROR-LOCATOR POLYNOMIAL a(X). .63 4.3.1 Method l:lterative Algorithm Fora(X) . ..64 4.3.1.1 Berlekamp’s Algorithm Presentation . ...65 4.3.1.2 Euclidean Division Algorithm Presentation . 67 4.3.2 Method 2: Linear Recursion Method for a(X) . .69 4.4 ERROR LOCATIONS \ . .. o. .71 4.4.1 Method l: Chien Search . ... ...71 4.4.2 Method 2: Explicit Factorization . , . ...73 4.5 ERROR VALUES ye. ...’... ..?3 4.5.1 . Method I: Direct Solution . ...73 4.5.2 Method 2: Error Evaluator Polynomials . ...75 4.5.2.1 ‘ Greatest Common Divisor Polynomial . ...75 4.5.2.2 - Hardware Error Evaluator Polynomial . ...76 4.6 DECODEDCODEWORDC(X)’ . ...77 4.7 SUMMARY . ...78 5 SYMBOLERASINGANDREED-SOLOMON CODING . ...80 5.1 RSCODING USINGSYMBOLERASURE . ...80 5.2 RSENCODINGUSINGSYMBOLERASURE . .,.84 - 5.3 RSDECODINGUSINGSYMBOLERASURE . ..85 5.3.1 Erasure Determination . ...85 5.3.2 Syndrome Components Si . ...86 5.3.3 Modified Syndrome Components S(’ . ...86 5.3.4 Error-Locator Coefficients ai . ...87 5.3.5 Error Locations xi . ...88 5.3.6 Error Values ye . ...88 5.3.7 Decoded Code WordC(X)’’ . ...89 5.3.8 Determining. t~ina Svmbol Erasure System . ...90 5.4 SUMMARY . .-. ...92 .—— –ii-- ‘\isc-21s34 APPENDIX A: RSHARDWARE ENCODING DESIGN . ..93 A.1 (15,9) RSSHIFT -REGISTE RENCODERCIRCUIT . ...93 A.2 OTHER RSSHl~REGISTER ENCODER CIRCUITS. ...97 APPENDIX B: RSHARDWARE DECODING DESIGN . ..99 B.1 BCHSHl~REGISTER SYNDROME CIRCUITS . ...100 B.2 GENERAL HARDWARE CIRCUITS FOR Ui. aj=ak . ...101 B.3 DIRECT METHOD HARDWARE FOR O(X) . ...103 B.3. I Previously Calculated Determinants . ..103 B.3.2 Reduced o(X) Computations .~. ...105 B.4 HIGH DATA RATE RSDECODINGDESIGN . ..105 APPENDIXC:MATRICESANDRSCODING . ...110 C.1 - RSENCODINGUSINGMATRICES . ...110 C.2 RSDECODINGUSINGMATRICES. ...113 APPENDIXD:AGENERAL MATHEMATICALOVERVIEWOF RECODING ..,.. ...117 RE~RENCES . ...122 ADDITIONALRECOMMENDEDREADING . ...123 ... --Ill– llI:;c-?f:;3-l TABLES Table . ... .,O$ ,oc, ,m# ................................. Page 1 .4.2-1 GF(16) ELEMENTS WITH F(X)= X4+X+ I USING (X(X)=X . ...19 1 .4.3-1 EQUIVALENT ELEMENT REPRESENTATIONS .......................”.....21 1.4.4-1 GF(16)ELEMENTS WITH F(X)=X4+ X + 1 USING CX(X)=X2 . ...25 1 .4.5-1 ADDITION/SUBTRACTION TABLE USED IN GF(24) . ... ..27 1.4.5-2 GF(16)ADDlT10N/SUBTRACTIQNTABLEUSED . ..
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