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The art of error correcting coding - Moreloz R.H.

Moreloz R.H. The art of error correcting coding - Wiley publishing , 2002. - 232 p.
ISBN 0471-49581-6
Download (direct link): artoferrorcorrecting2002.pdf
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bit error probability, AWGN channel, 57
Chemoff, 19 Hamming, 11
nonbinary case, 12 RS decoder, 72 union AWGN channel, 16 convolutional code, 84 multilevel modulation code for UEP, 186 Burst error correcting capability, 113
Chase algorithm, 129 soft output, 156 correction factor, 157 scaling factor, 157 Chien search, 55 Code
self-dual, 7, 29 Coded modulation bit-interleaved (BICM), 189 main idea, 2 MCM, 2, 173, 180 multistage decoding, 182 parallel decoding, 185 unequal error protection, 183 Pragmatic TCM symbol transformation, 177 two-stage decoding, 176 TCM, 2, 173, 174 example modified state diagram, 176 MLD decoding, 175 turbo trellis(TTCM), 192 Coding gain, 2, 16 asymptotic, 173, 187 Complementary error function (erfc), 2 Concatenated code, 115 Concatenated coding, 2 Conjugate elements, 43 Constellation, 170 Construction X3, 109 Convolutional code, 2 block code obtained from, 79 complete weight enumerator sequence, 83 constraint length, 74
218
THE ART OF ERROR CORRECTING CODING
defined, 3
direct-truncation, 80 finite-state machine, 73 generator sequences, 75 polynomial generator matrix, 77 recursive systematic (RSC code), 78 RSC code, 144 state diagram, 74 tail-biting, 80
union bound over BSC and AWGN channels, 84
weight distribution block codes from, 81-84 weight enumerating sequence, 82 zero-tail code, 79 Correlation discrepancy, 133 Coset, 104 decomposition, 104, 109, 118 leader, 10
representative, 118, 140 CRC code, 37
popular polynomials, 38 Cycle set, 43 Cyclic code defined, 33
encoding by division by g{x), 36
extended, 57
general decoder, 39
MLS code, 37
RM code, 29
shortened, 37
syndrome decoding, error-trapping, 39 zeros of, 34 Cyclic shift, 33 Cyclotomic coset, 43
Decoding BCH codes general, 48 BCJR algorithm, 137 Belief propagation, 162 Berlekamp-Massey algorithm, 47, 49-52 Chase algorithm, 129 soft output, 156 Chien search, 55 Depth, 88
Euclidean algorithm, 47, 53-54 Forney algorithm for RS codes, 62 GMD algorithm, 132 Log-MAP algorithm, 139 look-up table, 10 MAP algorithm, 137 Massey algorithm, 63 Max-Log-MAP algorithm, 140 MLD, 86, 175
Modified Forney algorithm, 68 ordered statistics algorithm, 131 soft-output, 140 Parallel for multilevel codes, 185 PGZ algorithm, 47, 52-53 SOVA algorithm, 134 Sudan algorithm, 67, 134 SW-SOVA algorithm, 136 two-stage, 115, 176 Viterbi algorithm, 85-94 with standard array, 8 Decomposable code, 107, 117 Direct-truncation code, 80 Discrepancy Berlelamp-Massey algorithm, 49 Disjunctive normal form, 27 Distance
designed of BCH code, 44 free, 79 Hamming, 4 minimum Hamming, 4 minimum squared Euclidean, 173 squared Euclidean, 15 Dual code, 6 example, 7 of cyclic code, 37 of RM code, 28
Encoding
non-systematic, 35
recursive with parity-check matrix, 36 systematic, 16, 35 with generator matrix, 8 with parity-check matrix, 8 Erasure, 55 value, 67
Erasure correction for binary linear codes, 55
Erasure locator polynomial, 67
Error
positions, 47 values, 47 Error bursts, 2
Error correcting capability, 5 Error correcting code as subset, 4 defined,3
minimum Hamming distance, 4 Error evaluator polynomial, 62 Error locator polynomial, 47 Error polynomial, 46 Error propagation, 186 Euclidean algorithm, 47, 53-54
polynomials same up to a constant, 66 Euclidean geometry (EG) code, 45
INDEX
219
Extended code, 103
Factoring polynomials in GF(2m) is hard, 55 Field, 40 Galois, 40 arithmetic, 41 element order, 43 representations, 41 Finding factors of a;2 1 + 1, 43 Finite geometry, 28 Flat Rayleigh fading channel bound, 18 model, 17 Forney algorithm, 62 Fourier transform
BCH decoding with, 67 Free distance, 79
Gallager code, 160 Galois field, 40
Generalized concatenated code, 117 array codes, 113 Generator matrix, 6 Generator polynomial, 34 of BCH code, 44 of RS code, 62 GMD decoding, 132 Golay code, 25
Greatest common divisor (GCD), 53
Hamming code, 23 shortened (71,64,3) code, 37 Hamming space defined, 4 distance, 4 sphere, 5 Hard-decision decoding, 16 general structure, 20
Incidence vector, 29 Inner code, 110 Interleaver, 152 block, 111 convolutional, 116 cyclic, 114
Ramsey, 111, 116, 152 random, 152 S-random, 152 Irreducible factors, 34 Irregular LDPC code record performance, 3 Iterative belief propagation algorithm message passing, 163 Iterative belief propagation decoding, 162
Iterative bit-flip decoding, 161 Iterative decoding convergence, 152
Key equation, 47
LDPC code, 159
error detection capability, 166 Likelihood, 85 Linear code
as vector subspace, 6 Linear feedback shift-register (LFSR), 49 List decoding, 134 Log and antilog tables, 42 Log-likelihood metric, 122 Log-likelihood ratio (LLR), 145 extrinsic, 146 Log-MAP algorithm, 139 Low-density parity-check code, 159
Mac Williams identity, 56 MAP algorithm, 137 Massey algorithm, 63 Matrix
generator and parity-check, 6 Vandermonde, 45 Max-Log-MAP algorithm, 140 Maximum-a-posteriori probability, 137 Maximum-distance-separable (MDS) code, 62 weight distribution, 71 Maximum-length sequence (MLS) code, 37 Meggit decoder, 39 Message passing, 163 Metric log-likelihood, 122 Metric normalization, 90 Minimal polynomial, 43 Minimum Hamming distance, 4 Minimum squared Euclidean distance, 173 MLD decoding, 175 defined,15 Viterbi algorithm, 86 Modified Forney syndrome, 67 Modified syndrome polynomial, 67 Modulation as mapping, 170 Monte Carlo integration, 19 MSED, 173 Multilevel coding, 180 Multilevel modulation code definition, 181 Multistage decoding, 182
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