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. 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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 . ..