DOCSLIB.ORG

Introduction to Coding Theory Ron M

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Coding Theory Ron M Cambridge University Press 978-0-521-84504-5 - Introduction to Coding Theory Ron M. Roth Index More information Index adjacency matrix block code, 5 of digraphs, 486 Blokh–Zyablov bound, 393 of graphs, 398 bound Lee, 325 BCH, 253 alphabet, 3 Bhattacharyya, 21, 25, 493 alternant code, 70, 157, 179 Blokh–Zyablov, 393 decoding of, 197, 204 Carlitz–Uchiyama, 179 Lee-metric, 306 Chernoff, 139 list, 280, 328 decoding-radius, 290 designed minimum distance of, 157, Elias, 108 250 Lee-metric, 332 dual code of, 175, 180 Gilbert–Varshamov, 97, 137, 176, 181, Lee-metric, 302 393 list decoding of, 280, 328 asymptotic, 107, 372 over Z, 328 Lee-metric, 320, 330 aperiodic irreducible digraph, 455 Griesmer, 120, 136 aperiodic irreducible matrix, 445 Hamming, see bound, sphere-packing arc (in projective geometry), 361 Hartmann–Tzeng, 265 complete, 363 Johnson, 107, 128, 139, 289 autocorrelation Lee-metric, 330 of Legendre sequences, 80 linear programming, 103, 110, 138 of maximal-length sequences, 87 MDS code length, 338 AWGN channel, 17 MRRW, 110 Plotkin, 37, 127, 131, 139, 294 basis Lee-metric, 326, 330 complementary, 85 Reiger, 122 dual, 85 Roos, 265 see normal, 240 Singleton, Singleton bound BCH bound, 253 sphere-covering, 123 BCH code, 162, 181, 244, 250 sphere-packing, 95, 122, 136 consecutive root sequence of, 163 asymptotic, 107 decoding of, see alternant code, decod- Lee-metric, 318, 330 ing of union, 137 designed minimum distance of, 163, Zyablov, 373, 392, 413, 422, 438, 440 250 burst, 45, 122, 137, 257 excess root of, 251 root of, 163, 250 cap (in projective geometry), 47 Berlekamp code, 314, 330 capacity, 10, 16, 24, 110 Berlekamp–Massey algorithm, 200, 217 Carlitz–Uchiyama bound, 179 Bhattacharyya bound, 21, 25, 493 catastrophic error propagation, 496 bi-connection, 454 Cauchy matrix, 168, 336, 356, 362 binary erasure channel, 15 Cayley graph, 406, 447 binary symmetric channel (BSC), 4, 450 channel, 1 bipartite graph, 362, 398 additive, 5 bit-shift error, 327 AWGN, 17 Blahut’s algorithm, 217 binary symmetric (BSC), 4, 450 559 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-84504-5 - Introduction to Coding Theory Ron M. Roth Index More information 560 Index discrete memoryless (DMC), 19, 23, 46, negacyclic, 323, 330 142, 466 outer (in concatenated codes), 154, 366 erasure, 15, 25, 126, 134, 391, 514 parity, 27 Gaussian (AWGN), 17 perfect, 96, 137, 256 non-symmetric, 20, 46 Lee-metric, 319, 330 probabilistic, 3 Preparata, 328 symmetric, 4, 19, 24, 32, 110, 113, 117, product, 44, 178 125, 133, 145, 378, 390, 394, 424, punctured, 36 450, 467, 490 rate, 5 character redundancy, 27 of Abelian groups, 91, 447 Reed–Muller, see Reed–Muller code additive, 85, 99, 447 Reed–Solomon, see RS code multiplicative, 78 repetition, 28 quadratic, 80 self-dual, 31 trivial, 79 shortened, 40 characteristic (of fields), 62 simplex, 41, 120 check node, 449 weight distribution of, 99 check polynomial, 246 size, 5 Chernoff bound, 139 spectral-null, 329 Chien search, 186, 215, 285 Spielman, 451 circulant matrix, 325, 330, 356, 362 trellis, 460 code, 5 decoding of, 466 algebraic-geometry, 138 encoding of, 464 almost-MDS (AMDS), 363 free distance of, 463 alternant, see alternant code turbo, 519 array, 353 Wozencraft, 375 BCH, see BCH code Wyner–Ash, 512 Berlekamp, 314, 330 codeword, 3, 460 block, 5 communication system, 1 concatenated, see concatenated code companion matrix, 73, 383, 510 constant-weight, 121, 139, 352 complementary basis, 85 convolutional, see convolutional code concatenated code, 154, 172, 178, 408, 420 cyclic, see cyclic code decoding of, 178, 371, 396, 422 Delsarte–Goethals, 328 dual code of, 383 dimension, 5 linearly-, 154, 367 double-error-correcting, 70, 161 concave function, 9, 23 dual, see dual code conjugate element equidistant, 128 in the complex field, 325 equivalence, 29, 47 in cyclotomic extension fields, 241 generalized Reed–Solomon, see GRS in finite extension fields, 218 code conjugate transpose (of complex matrices), Golay, 96, 136, 255 326 Goppa, 182, 389 convex function, 9 graph, see graph code convolutional code, 477 Gray, 321, 328 constraint length of encoders of, 483 group, 37, 299 decoding of, 485, 519 Hamming, see Hamming code encoding of, 479 inner (in concatenated codes), 154, 366 free distance of, 478, 512 Justesen, 376 generator matrix of, 477 Kerdock, 328 coset LDPC, 362, 450 leader, 34 length, 5 of linear codes, 34 lengthening of, 123 of subgroups, 522 linear, see linear code cover (of arrays), 354, 362 low-density parity-check, 362, 450 covering radius, 123, 137 maximal, 123 of GRS codes, 166 maximum distance separable (MDS), of MDS codes, 166 see MDS code crisscross error, 362 minimum distance, 6 cycle (in graphs), 397 near-MDS (NMDS), 363 in digraphs, 454 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-84504-5 - Introduction to Coding Theory Ron M. Roth Index More information Index 561 cyclic code, 242 irreducible component of, 455, 501 dual code of, 247 irreducible sink of, 455, 501 encoding of, 245 lossless, 456 Hamming, 244, 254, 264 period of, 455, 502 of length q over GF(q), 257, 265 primitive irreducible, 455 of length q+1 over GF(q), 262, 265 regular, 459 repeated-root, 265 sequence set of, 457 root of, 248 strongly-connected, 454, 519 shortened, 388 tag in, 459 cyclotomic coset, 230 trellis diagram of, 458 cyclotomic extension field, 240 dimension (of codes), 5, 26 direct product (of matrices), 45, 428 dB, 18 directed graph, see digraph (and labeled di- decoding, 7 graph) of alternant codes, see alternant code, discrete logarithm, 59, 81, 91 decoding of discrete memoryless channel (DMC), 19, 23, of BCH codes, see alternant code, de- 46, 142, 466 coding of distance complexity, 48 cover, 354, 362 of concatenated codes, 178, 371, 396, free, 463, 478, 512 422 in graphs, 397 of convolutional codes, 485, 519 Hamming, 6 error probability, 7, 110, 132, 133, 471, Kullback–Leibler, 111 485, 491, 519 Lee, 299 generalized minimum distance (GMD), rank, 19, 353, 361 178, 371, 396, 422 divergence, 111 of graph codes, 414 dual basis, 85 of GRS codes, see GRS decoding dual code, 30 of Hamming codes, 35 of alternant codes, 175, 180 hard-decision, 18 of concatenated codes, 383 iterative, 414, 451 of cyclic codes, 247 of linear codes, 33 of extended GRS codes, 163 list, see list decoding of GRS codes, 148 of GRS codes, see GRS list decoding of Hamming codes, 41 maximum a posteriori,8 of MDS codes, 119 maximum-likelihood, 8, 32, 140, 466 of RS codes, 257 misdetection probability, 22, 125, 132, self-, 31 140, 440 of subfield sub-codes, 175 nearest-codeword, 9, 33 dual linear programming problem, 138 sequential, 520 dynamic programming, 519 soft-decision, 18 standard array, 33 edge (in graphs), 396 syndrome, 34 cut, 397 of trellis codes, 466 in digraphs, 453 degree (of extension fields), 57 Elias bound, 108 degree (of polynomials), 51 Lee-metric, 332 of bivariate polynomials, 268 encoder, 3 degree (of vertices in graphs), 396 entropy function derivative, 65, 87, 194, 300, 322 binary, 9 Hasse (or hyper-), 87, 276, 310, 329 q-ary, 24, 105 designed minimum distance, 157, 163, 250 erasure, 15 diameter (of graphs), 397 burst, 45, 257 digraph (and labeled digraph), 400, 453 channel, 15, 25, 126, 134, 391, 514 adjacency matrix of, 486 error, 12 anticipation of, 503 bit-shift, 327 aperiodic irreducible, 455 burst, 45, 122, 137, 257 controllable, 454, 519 correction, 12 deterministic, 456 crisscross, 362 induced, 453 detection, 13 irreducible, 454, 519 evaluator polynomial, 186 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-84504-5 - Introduction to Coding Theory Ron M. Roth Index More information 562 Index exponent, 143, 381, 393, 425 generalized minimum distance (GMD) de- location, 5 coder, 178, 371, 396, 422 locator polynomial, 185 generalized Reed–Solomon code, see GRS Lee-metric, 307 code locator ratio, 307 generator matrix (of convolutional codes), peak-shift, 327 477 synchronization, 327 catastrophic, 497 value, 5 LFSM realization of, 510 word, 5 systematic, 477, 517 error-evaluator polynomial, 186 generator matrix (of linear codes), 27 error-locator polynomial, 185 systematic, 29 Lee-metric, 307 generator polynomial error-locator ratio, 307 of cyclic codes, 245 Euclid’s algorithm of negacyclic codes, 323 for integers, 50, 501, 524 of RS codes, 152 for polynomials, 52, 71, 90, 191, 215, Gilbert–Varshamov bound, 97, 137, 176, 309 181, 393 Euler–Fermat Theorem, 526 asymptotic, 107, 372 Euler function, 62, 229, 449, 522 Lee-metric, 320, 330 expander (graph), 404 GMD decoder, 178, 371, 396, 422 exponent (of polynomials), 206, 227, 247 Golay code, 96, 136, 255 extension field, 57, 218 Goppa code, 182, 389 arithmetic in, 59, 74, 90 graph conjugate element in, 218, 241 adjacency matrix of, 398 cyclotomic, 240 bipartite, 362, 398 transfer matrix of, 399 Cayley, 406, 447 factorization of polynomials, 56, 90 code, see graph code Fano’s algorithm, 520 connected, 397 Fermat’s Little Theorem, 526 directed, see digraph (and labeled di- field, 522 graph) characteristic of, 62 edge cut in, 397 extension, see extension field expander, 404 finite, see finite field Hamming, 427 Galois, see finite field hyper-, 445 prime, 50 incidence matrix of, 399 of rational functions, 268, 476 induced, 397 splitting, 65 isomorphism, 47 finite field, 50, 218, 240 labeled directed, see digraph (and la- characteristic of, 62 beled digraph) see extension field of, extension field Laplace matrix of, 399 isomorphism,
Load more

Recommended publications
  • Ts 136 212 V11.1.0 (2013-02)
    ETSI TS 136 212 V11.1.0 (2013-02) Technical Specification LTE; Evolved Universal Terrestrial Radio Access (E-UTRA); Multiplexing and channel coding (3GPP TS 36.212 version 11.1.0 Release 11) 3GPP TS 36.212 version 11.1.0 Release 11 1 ETSI TS 136 212 V11.1.0 (2013-02) Reference RTS/TSGR-0136212vb10 Keywords LTE ETSI 650 Route des Lucioles F-06921 Sophia Antipolis Cedex - FRANCE Tel.: +33 4 92 94 42 00 Fax: +33 4 93 65 47 16 Siret N° 348 623 562 00017 - NAF 742 C Association à but non lucratif enregistrée à la Sous-Préfecture de Grasse (06) N° 7803/88 Important notice Individual copies of the present document can be downloaded from: http://www.etsi.org The present document may be made available in more than one electronic version or in print. In any case of existing or perceived difference in contents between such versions, the reference version is the Portable Document Format (PDF). In case of dispute, the reference shall be the printing on ETSI printers of the PDF version kept on a specific network drive within ETSI Secretariat. Users of the present document should be aware that the document may be subject to revision or change of status. Information on the current status of this and other ETSI documents is available at http://portal.etsi.org/tb/status/status.asp If you find errors in the present document, please send your comment to one of the following services: http://portal.etsi.org/chaircor/ETSI_support.asp Copyright Notification No part may be reproduced except as authorized by written permission.
    [Show full text]
  • Tm Synchronization and Channel Coding—Summary of Concept and Rationale
    Report Concerning Space Data System Standards TM SYNCHRONIZATION AND CHANNEL CODING— SUMMARY OF CONCEPT AND RATIONALE INFORMATIONAL REPORT CCSDS 130.1-G-3 GREEN BOOK June 2020 Report Concerning Space Data System Standards TM SYNCHRONIZATION AND CHANNEL CODING— SUMMARY OF CONCEPT AND RATIONALE INFORMATIONAL REPORT CCSDS 130.1-G-3 GREEN BOOK June 2020 TM SYNCHRONIZATION AND CHANNEL CODING—SUMMARY OF CONCEPT AND RATIONALE AUTHORITY Issue: Informational Report, Issue 3 Date: June 2020 Location: Washington, DC, USA This document has been approved for publication by the Management Council of the Consultative Committee for Space Data Systems (CCSDS) and reflects the consensus of technical panel experts from CCSDS Member Agencies. The procedure for review and authorization of CCSDS Reports is detailed in Organization and Processes for the Consultative Committee for Space Data Systems (CCSDS A02.1-Y-4). This document is published and maintained by: CCSDS Secretariat National Aeronautics and Space Administration Washington, DC, USA Email: [email protected] CCSDS 130.1-G-3 Page i June 2020 TM SYNCHRONIZATION AND CHANNEL CODING—SUMMARY OF CONCEPT AND RATIONALE FOREWORD This document is a CCSDS Report that contains background and explanatory material to support the CCSDS Recommended Standard, TM Synchronization and Channel Coding (reference [3]). Through the process of normal evolution, it is expected that expansion, deletion, or modification of this document may occur. This Report is therefore subject to CCSDS document management and change control procedures, which are defined in Organization and Processes for the Consultative Committee for Space Data Systems (CCSDS A02.1-Y-4). Current versions of CCSDS documents are maintained at the CCSDS Web site: http://www.ccsds.org/ Questions relating to the contents or status of this document should be sent to the CCSDS Secretariat at the email address indicated on page i.
    [Show full text]
  • Symbol Index
    Cambridge University Press 978-0-521-13170-4 - Fundamentals of Error-Correcting Codes W. Cary Huffman and Vera Pless Index More information Symbol index ⊥,5,275 Aq (n, d,w), 60 n ⊥H ,7 A (F), 517 , 165 Aut(P, B), 292 , 533 Bi (C), 75 a , p , 219 B(n d), 48 (g(x)), 125 Bq (n, d), 48 g(x) , 125, 481 B(G), 564 x, y ,7 B(t, m), 439 x, y T , 383 C1 ⊕ C2,18 αq (δ), 89, 541 C1 C2, 368 Aut(C), 26, 384 Ci /C j , 353 AutPr(C), 28 Ci ⊥C j , 353 (L, G), 521 C∗,13 μ, 187 C| , 116 Fq (n, k), 567 Cc, 145 (q), 426 C,14 2, 423 C(L), 565 ⊥ 24, 429 C ,5,469 ∗ ⊥ , 424 C H ,7 C ⊥ ( ), 427 C T , 384 C 4( ), 503 Cq (A), 308 x, y ( ), 196 Cs , 114, 122 λ i , 293 CT ,14 λ j i , 295 CT ,16 λ i (x), 609 C(X , P, D), 535 μ a , 138 d2m , 367 ν C ( ), 465 d free, 563 ν P ( ), 584 dr (C), 283 ν s , s ( i j ), 584 Dq (A), 309 π (n) C, i (x), 608, 610 d( x), 65 ρ(C), 50, 432 d(x, y), 7 , ρBCH(t, m), 440 dE (x y), 470 , ρRM(r, m), 437 dH (x y), 470 , σp, 111 dL (x y), 470 σ (x), 181 deg f (x), 101 σ (μ)(x), 187 det , 423 Dext φs , 164 , 249 φs : Fq [I] → I, 164 e7, 367 ω(x), 190 e8, 367 A , 309 E8, 428 T A ,4 Eb, 577 E Ai (C), 8, 252 n, 209 A(n, d), 48 Es , 573 Aq (n, d), 48 evP , 534 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-13170-4 - Fundamentals of Error-Correcting Codes W.
    [Show full text]
  • Ioag Infusion Plans and Roadmaps
    IOAG ROADMAP DRAFT IOAG INFUSION PLANS AND ROADMAPS (this version contains all IOAG agency(s) input) (Draft IOAG-8 Version – July 2005) (Draft IOAG-8 Version, Revision 1 – August 2005) IOAG ROADMAP This document is uncontrolled when printed. Please check the IOAG web site at http://www.ioag.org prior to use to ensure this is the latest version. IOAG ROADMAP TABLE OF CONTENTS 1 – INTRODUCTION .................................................................................................................1 2 – STRUCTURE AND MANAGEMENT OF THE PRESENT DOCUMENT...........................1 2-1 STRUCTURE ..................................................................................................................1 2-2 DOCUMENT MANAGEMENT..........................................................................................2 3 – IOAG STRATEGY PLAN....................................................................................................2 3-1 IOAG OBJECTIVES ........................................................................................................2 3-2 IOAG AREAS OF INTEREST ..........................................................................................3 4 – IOAG INTEROPERABILITY ITEMS...................................................................................6 5 – AGENCIES’ INFUSION PLANS AND ROADMAPS..........................................................7 ANNEX-1 – ASI INFUSION PLAN AND ROADMAP ...........................................................1-1 ANNEX-2 – CNES INFUSION PLAN AND ROADMAP
    [Show full text]
  • Reed-Solomon Codes
    Foreword This chapter is based on lecture notes from coding theory courses taught by Venkatesan Gu- ruswami at University at Washington and CMU; by Atri Rudra at University at Buffalo, SUNY and by Madhu Sudan at MIT. This version is dated May 8, 2015. For the latest version, please go to http://www.cse.buffalo.edu/ atri/courses/coding-theory/book/ The material in this chapter is supported in part by the National Science Foundation under CAREER grant CCF-0844796. Any opinions, findings and conclusions or recomendations ex- pressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF). ©Venkatesan Guruswami, Atri Rudra, Madhu Sudan, 2013. This work is licensed under the Creative Commons Attribution-NonCommercial- NoDerivs 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA. Chapter 5 The Greatest Code of Them All: Reed-Solomon Codes In this chapter, we will study the Reed-Solomon codes. Reed-Solomoncodeshavebeenstudied a lot in coding theory. These codes are optimal in the sense that they meet the Singleton bound (Theorem 4.3.1). We would like to emphasize that these codes meet the Singleton bound not just asymptotically in terms of rate and relative distance but also in terms of the dimension, block length and distance. As if this were not enough, Reed-Solomon codes turn out to be more versatile: they have many applications outside of coding theory.
    [Show full text]
  • Short Low-Rate Non-Binary Turbo Codes Gianluigi Liva, Bal´Azs Matuz, Enrico Paolini, Marco Chiani
    Short Low-Rate Non-Binary Turbo Codes Gianluigi Liva, Bal´azs Matuz, Enrico Paolini, Marco Chiani Abstract—A serial concatenation of an outer non-binary turbo decoding thresholds lie within 0.5 dB from the Shannon code with different inner binary codes is introduced and an- limit in the coherent case, for a wide range of coding rates 1 alyzed. The turbo code is based on memory- time-variant (1/3 ≤ R ≤ 1/96). Remarkably, a similar result is achieved recursive convolutional codes over high order fields. The resulting codes possess low rates and capacity-approaching performance, in the noncoherent detection framework as well. thus representing an appealing solution for spread spectrum We then focus on the specific case where the inner code is communications. The performance of the scheme is investigated either an order-q Hadamard code or a first order length-q Reed- on the additive white Gaussian noise channel with coherent and Muller (RM) code, due to their simple fast Hadamard trans- noncoherent detection via density evolution analysis. The pro- form (FHT)-based decoding algorithms. The proposed scheme posed codes compare favorably w.r.t. other low rate constructions in terms of complexity/performance trade-off. Low error floors can be thus seen either as (i) a serial concatenation of an outer and performances close to the sphere packing bound are achieved Fq-based turbo code with an inner Hadamard/RM code with down to small block sizes (k = 192 information bits). antipodal signalling, or (ii) as a coded modulation Fq-based turbo/LDPC code with q-ary (bi-) orthogonal modulation.
    [Show full text]
  • Tm Synchronization and Channel Coding
    Recommendation for Space Data System Standards TM SYNCHRONIZATION AND CHANNEL CODING RECOMMENDED STANDARD CCSDS 131.0-B-3 BLUE BOOK September 2017 Recommendation for Space Data System Standards TM SYNCHRONIZATION AND CHANNEL CODING RECOMMENDED STANDARD CCSDS 131.0-B-3 BLUE BOOK September 2017 CCSDS RECOMMENDED STANDARD FOR TM SYNCHRONIZATION AND CHANNEL CODING AUTHORITY Issue: Recommended Standard, Issue 3 Date: September 2017 Location: Washington, DC, USA This document has been approved for publication by the Management Council of the Consultative Committee for Space Data Systems (CCSDS) and represents the consensus technical agreement of the participating CCSDS Member Agencies. The procedure for review and authorization of CCSDS documents is detailed in Organization and Processes for the Consultative Committee for Space Data Systems (CCSDS A02.1-Y-4), and the record of Agency participation in the authorization of this document can be obtained from the CCSDS Secretariat at the e-mail address below. This document is published and maintained by: CCSDS Secretariat National Aeronautics and Space Administration Washington, DC, USA E-mail: [email protected] CCSDS 131.0-B-3 Page i September 2017 CCSDS RECOMMENDED STANDARD FOR TM SYNCHRONIZATION AND CHANNEL CODING STATEMENT OF INTENT The Consultative Committee for Space Data Systems (CCSDS) is an organization officially established by the management of its members. The Committee meets periodically to address data systems problems that are common to all participants, and to formulate sound technical solutions to these problems. Inasmuch as participation in the CCSDS is completely voluntary, the results of Committee actions are termed Recommended Standards and are not considered binding on any Agency.
    [Show full text]
  • Download Special Issue
    Wireless Communications and Mobile Computing Error Control Codes for Next-Generation Communication Systems: Opportunities and Challenges Lead Guest Editor: Zesong Fei Guest Editors: Jinhong Yuan and Qin Huang Error Control Codes for Next-Generation Communication Systems: Opportunities and Challenges Wireless Communications and Mobile Computing Error Control Codes for Next-Generation Communication Systems: Opportunities and Challenges Lead Guest Editor: Zesong Fei Guest Editors: Jinhong Yuan and Qin Huang Copyright © 2018 Hindawi. All rights reserved. This is a special issue published in “Wireless Communications and Mobile Computing.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, pro- vided the original work is properly cited. Editorial Board Javier Aguiar, Spain Oscar Esparza, Spain Maode Ma, Singapore Ghufran Ahmed, Pakistan Maria Fazio, Italy Imadeldin Mahgoub, USA Wessam Ajib, Canada Mauro Femminella, Italy Pietro Manzoni, Spain Muhammad Alam, China Manuel Fernandez-Veiga, Spain Álvaro Marco, Spain Eva Antonino-Daviu, Spain Gianluigi Ferrari, Italy Gustavo Marfia, Italy Shlomi Arnon, Israel Ilario Filippini, Italy Francisco J. Martinez, Spain Leyre Azpilicueta, Mexico Jesus Fontecha, Spain Davide Mattera, Italy Paolo Barsocchi, Italy Luca Foschini, Italy Michael McGuire, Canada Alessandro Bazzi, Italy A. G. Fragkiadakis, Greece Nathalie Mitton, France Zdenek Becvar, Czech Republic Sabrina Gaito, Italy Klaus Moessner, UK Francesco Benedetto, Italy Óscar García, Spain Antonella Molinaro, Italy Olivier Berder, France Manuel García Sánchez, Spain Simone Morosi, Italy Ana M. Bernardos, Spain L. J. García Villalba, Spain Kumudu S. Munasinghe, Australia Mauro Biagi, Italy José A. García-Naya, Spain Enrico Natalizio, France Dario Bruneo, Italy Miguel Garcia-Pineda, Spain Keivan Navaie, UK Jun Cai, Canada A.-J.
    [Show full text]
  • Scribe Notes
    6.440 Essential Coding Theory Feb 19, 2008 Lecture 4 Lecturer: Madhu Sudan Scribe: Ning Xie Today we are going to discuss limitations of codes. More specifically, we will see rate upper bounds of codes, including Singleton bound, Hamming bound, Plotkin bound, Elias bound and Johnson bound. 1 Review of last lecture n k Let C Σ be an error correcting code. We say C is an (n, k, d)q code if Σ = q, C q and ∆(C) d, ⊆ | | | | ≥ ≥ where ∆(C) denotes the minimum distance of C. We write C as [n, k, d]q code if furthermore the code is a F k linear subspace over q (i.e., C is a linear code). Define the rate of code C as R := n and relative distance d as δ := n . Usually we fix q and study the asymptotic behaviors of R and δ as n . Recall last time we gave an existence result, namely the Gilbert-Varshamov(GV)→ ∞ bound constructed by greedy codes (Varshamov bound corresponds to greedy linear codes). For q = 2, GV bound gives codes with k n log2 Vol(n, d 2). Asymptotically this shows the existence of codes with R 1 H(δ), which is similar≥ to− Shannon’s result.− Today we are going to see some upper bound results, that≥ is,− code beyond certain bounds does not exist. 2 Singleton bound Theorem 1 (Singleton bound) For any code with any alphabet size q, R + δ 1. ≤ Proof Let C Σn be a code with C Σ k. The main idea is to project the code C on to the first k 1 ⊆ | | ≥ | | n k−1 − coordinates.
    [Show full text]
  • Fundamentals of Error-Correcting Codes W
    Cambridge University Press 978-0-521-13170-4 - Fundamentals of Error-Correcting Codes W. Cary Huffman and Vera Pless Frontmatter More information Fundamentals of Error-Correcting Codes Fundamentals of Error-Correcting Codes is an in-depth introduction to coding theory from both an engineering and mathematical viewpoint. As well as covering classical topics, much coverage is included of recent techniques that until now could only be found in special- ist journals and book publications. Numerous exercises and examples and an accessible writing style make this a lucid and effective introduction to coding theory for advanced undergraduate and graduate students, researchers and engineers, whether approaching the subject from a mathematical, engineering, or computer science background. Professor W. Cary Huffman graduated with a PhD in mathematics from the California Institute of Technology in 1974. He taught at Dartmouth College and Union College until he joined the Department of Mathematics and Statistics at Loyola in 1978, serving as chair of the department from 1986 through 1992. He is an author of approximately 40 research papers in finite group theory, combinatorics, and coding theory, which have appeared in journals such as the Journal of Algebra, IEEE Transactions on Information Theory, and the Journal of Combinatorial Theory. Professor Vera Pless was an undergraduate at the University of Chicago and received her PhD from Northwestern in 1957. After ten years at the Air Force Cambridge Research Laboratory, she spent a few years at MIT’s project MAC. She joined the University of Illinois-Chicago’s Department of Mathematics, Statistics, and Computer Science as a full professor in 1975 and has been there ever since.
    [Show full text]
  • An Introduction to Coding Theory
    An introduction to coding theory Adrish Banerjee Department of Electrical Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh India Feb. 6, 2017 Lecture #7A: Bounds on the size of a code Adrish Banerjee Department of Electrical Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh India An introduction to coding theory Outline of the lecture Hamming bound Adrish Banerjee Department of Electrical Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh India An introduction to coding theory Outline of the lecture Hamming bound Perfect codes Adrish Banerjee Department of Electrical Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh India An introduction to coding theory Outline of the lecture Hamming bound Perfect codes Singleton bound Adrish Banerjee Department of Electrical Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh India An introduction to coding theory Outline of the lecture Hamming bound Perfect codes Singleton bound Maximum distance separable codes Adrish Banerjee Department of Electrical Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh India An introduction to coding theory Outline of the lecture Hamming bound Perfect codes Singleton bound Maximum distance separable codes Plotkin Bound Adrish Banerjee Department of Electrical Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh India An introduction to coding theory Outline of the lecture Hamming bound Perfect codes Singleton bound Maximum distance separable codes Plotkin Bound Gilbert-Varshamov bound Adrish Banerjee Department of Electrical Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh India An introduction to coding theory Bounds on the size of a code The basic problem is to find the largest code of a given length, n and minimum distance, d.
    [Show full text]
  • Optimal Codes and Entropy Extractors
    Department of Mathematics Ph.D. Programme in Mathematics XXIX cycle Optimal Codes and Entropy Extractors Alessio Meneghetti 2017 University of Trento Ph.D. Programme in Mathematics Doctoral thesis in Mathematics OPTIMAL CODES AND ENTROPY EXTRACTORS Alessio Meneghetti Advisor: Prof. Massimiliano Sala Head of the School: Prof. Valter Moretti Academic years: 2013{2016 XXIX cycle { January 2017 Acknowledgements I thank Dr. Eleonora Guerrini, whose Ph.D. Thesis inspired me and whose help was crucial for my research on Coding Theory. I also thank Dr. Alessandro Tomasi, without whose cooperation this work on Entropy Extraction would not have come in the present form. Finally, I sincerely thank Prof. Massimiliano Sala for his guidance and useful advice over the past few years. This research was funded by the Autonomous Province of Trento, Call \Grandi Progetti 2012", project On silicon quantum optics for quantum computing and secure communications - SiQuro. i Contents List of Figures......................v List of Tables....................... vii Introduction1 I Codes and Random Number Generation7 1 Introduction to Coding Theory9 1.1 Definition of systematic codes.............. 16 1.2 Puncturing, shortening and extending......... 18 2 Bounds on code parameters 27 2.1 Sphere packing bound.................. 29 2.2 Gilbert bound....................... 29 2.3 Varshamov bound.................... 30 2.4 Singleton bound..................... 31 2.5 Griesmer bound...................... 33 2.6 Plotkin bound....................... 34 2.7 Johnson bounds...................... 36 2.8 Elias bound........................ 40 2.9 Zinoviev-Litsyn-Laihonen bound............ 42 2.10 Bellini-Guerrini-Sala bounds............... 43 iii 3 Hadamard Matrices and Codes 49 3.1 Hadamard matrices.................... 50 3.2 Hadamard codes..................... 53 4 Introduction to Random Number Generation 59 4.1 Entropy extractors...................
    [Show full text]