Introduction to Coding Theory Ron M

Home , Johnson bound, Singleton bound, Turbo code

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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

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

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

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, 227 matching in, 362 prime, 50 oriented, 400 product of elements in, 77 Ramanujan, 409, 447 sum of powers of elements in, 77, 150 regular, 401 finite-state machine (FSM), 460 undirected simple, 396 see linear, linear finite-state machine graph code, 412 (LFSM) approaching the Singleton bound, 421 see formal derivative, derivative decoding of, 414 formal power series, 189, 237, 300, 474 error detection with, 440 bivariate, 487 generalized, 420 Forney’s algorithm, 195, 215 Gray code, 321, 328 Fourier transform, 81, 92, 217 Griesmer bound, 120, 136 free distance, 463, 478, 512 group, 521 Frobenius mapping, 64 Abelian, 521 code, 37, 299 Galois field, see finite field commutative, 521 Gaussian (AWGN) channel, 17 cyclic, 521 Gaussian elimination, 189, 274, 296 factor, 525 Gaussian noise, 17 projective special linear, 409

Index 563

quotient, 525 of bivariate polynomials, 276 special linear, 409 hyper-derivative, see Hasse derivative GRS code, 148 hyper-graph, 445 canonical generator matrix of, 167 canonical parity-check matrix of, 148 ideal, 522 code locator of, 148 incidence matrix, 399 decoding of, see GRS decoding infinite formal power series, 189, 237, 300, doubly-extended, 163, 335 474 dual code of, 148 bivariate, 487 encoding of, 152, 177, 216 integer (of fields), 63 extended, 150, 336 integer programming, 103, 137 dual code of, 163 integral domain, 522 Lee-metric, 304 interleaver, 44 list decoding of, see GRS list decoding interpolation, 76 narrow-sense, 150 bivariate, 270 normalized, 149 noisy, 151, 177 primitive, 149 rational, 211 singly-extended, 150, 335 invertible element (in a ring), 190 systematic generator matrix of, 167 irreducible matrix, 445 triply-extended, 165, 337 irreducible polynomial, 54, 90 GRS decoding, 184 enumeration of, 225 Berlekamp–Massey, 200, 217 bivariate interpolation, 269 Jacobi’s Four Square Theorem, 410 Blahut’s time-domain, 217 Jensen’s inequality, 23 Chien search in, 186, 215 Johnson bound, 107, 128, 139, 289 erasure, 207, 216 Lee-metric, 330 with Euclid’s algorithm, 191, 215 Justesen code, 376 extended, 210, 216 key equation, 186, 215 Forney’s algorithm for, 195, 215 Lee-metric, 308 key equation of, 186, 215 Koetter–Vardy algorithm, 282, 296 Lee-metric, 308 König’s Theorem, 362 Lee-metric, 312 Krawtchouk polynomial, 103, 124, 137 list, see GRS list decoding Kronecker product (of matrices), 45, 428 Peterson–Gorenstein–Zierler, 189, 215 Kullback–Leibler distance, 111 singly-extended, 210, 216 et al. Sugiyama (SKHN), 215 labeled directed graph, see digraph (and la- Welch–Berlekamp, 215, 217 beled digraph) equations of, 211, 215, 271, 291, 295 Lagrange’s Theorem, 525 GRS list decoding, 271 Laplace matrix, 399 Guruswami–Sudan, 278, 296 Latin square, 351, 361 Koetter–Vardy, 282, 296 Laurent series, 476, 507 radius, 271 Law of Large Numbers, 11 Sudan’s, 274, 296 Law of Quadratic Reciprocity, 411, 449 Guruswami–Sudan algorithm, 278, 296 LDPC code, 362, 450 Lee Hadamard matrix, 45, 49, 406, 436, 447 adjacency matrix, 325 Hamming bound, see sphere-packing bound distance, 299 Hamming code, 32, 47 weight, 299, 448 binary, 32 Legendre sequence, 80 cyclic, 244, 254, 264 autocorrelation of, 80 decoding of, 35 linear recurrence of, 239, 241 list, 291 Legendre symbol, 80, 447 dual code of, 41 length (of codes), 5 extended binary, 32 likelihood ratio, 20 list decoding of, 291 linear code, 26 weight distribution of, 101, 121 decoding of, 33 Hamming distance, 6 encoding of, 28 Hamming graph, 427 over rings, 43, 299, 328, 478 Hartmann–Tzeng bound, 265 linear-feedback shift register (LFSR), 198, Hasse derivative, 87, 310, 329 217

564 Index

maximal-length, 237 message-passing algorithm (MPA), 451 linear finite-state machine (LFSM), 472 metric, 6 catastrophic, 496, 516 cover, 354, 362 deterministic, 517 Hamming, 6 irreducible, 473, 505 Lee, 299 length–weight enumerator of, 487 rank, 19, 353, 361 lossless, 476, 508 minimal polynomial, 219, 241 observable, 496, 506, 519 minimum distance, 6 reduced, 496, 506 computation complexity of, 48, 177 response matrix of, 474, 519 designed, 157, 163, 250 linear programming, 137 Lee, 299 bound, 103, 110, 138 relative, 104 linearized polynomial, 83, 353 Möbius function, 224, 523 list decoding, 267 MRRW bound, 110 of alternant codes, 280, 328 multiplication circuit, 153 error probability, 267 multiplicative order of GRS codes, see GRS list decoding in fields, 51 of Hamming codes, 291 in groups, 521 radius, 267 in rings, 72, 206, 228 bound on, 290 mutual information, 23 log likelihood ratio, 20 low-density parity-check code, 362, 450 negacyclic code, 323, 330 Newton’s identities, 301, 328 node MacWilliams’ identities, 104, 124 check, 449 matching (in graphs), 362 message, 449 matrix variable, 449 adjacency, 398, 486 normal basis, 240 aperiodic irreducible, 445 normal rational curve, 362 Cauchy, 168, 336, 356, 362 circulant, 325, 330, 356, 362 order, see multiplicative order companion, 73, 383, 510 orthogonal array, 334, 361 complex conjugate transpose of, 326 direct product of, 45, 428 parity-check matrix, 29 generator, 27, 477 parity code, 27 incidence, 399 path (in graphs), 397 irreducible, 445 in digraphs, 453 Kronecker product of, 45, 428 peak-shift error, 327 Laplace, 399 perfect code, 96, 137, 256 Lee adjacency, 325 Lee-metric, 319, 330 parity-check, 29 period (of irreducible digraphs), 455, 502 primitive irreducible, 445 period (of sequences), 206, 237, 305 super-regular, 335 Perron–Frobenius Theorem, 445, 518 Sylvester-type Hadamard, 45, 49, 406, Peterson–Gorenstein–Zierler algorithm, 189, 436, 447 215 transfer, 399 phase-shift keying (PSK), 327 Vandermonde, 75 Plotkin bound, 37, 127, 131, 139, 294 inverse of, 166 Lee-metric, 326, 330 maximum-likelihood decoder, 8, 32, 140, 466 polynomial, 51 MDS code, 94, 119, 148, 164, 334 bivariate, 268 almost (AMDS), 363 characteristic, 73 bound on the length of, 338 check, 246 conjecture, 342, 363 degree, 51 dual code of, 119 error-evaluator, 186 near (NMDS), 363 error-locator, 185 over polynomial rings, 355 Lee-metric, 307 uniqueness of, 347 evaluation, 177, 216 weight distribution of, 104 exponent of, 206, 227, 247 MDS conjecture, 342, 363 factorization, 56, 90 Mersenne prime, 77 generator, 152, 245, 323 message node, 449 interpolation, 76, 151, 177, 211, 270

Index 565

irreducible, 54, 90 dual code of, 257 enumeration of, 225 encoding of, 152, 178 Krawtchouk, 103, 124, 137 extended, 152 linearized, 83, 353 generalized, see GRS code minimal, 219, 241 generator polynomial of, 152 monic, 52 root of, 152, 163, 250 multivariate, 177, 295 primitive, 228 self-dual code, 31 enumeration of, 229 sequence, 189 quadratic, 78, 88 Legendre, 80 root of, 59 autocorrelation of, 80 syndrome, 185 linear recurrence of, 239, 241 power series, 189, 237, 300, 474 linear-recurring, 198, 217, 237 bivariate, 487 M-, 237, 241 Prime Number Theorem, 411, 449 maximal-length, 237, 241 primitive element, 51, 61, 91 autocorrelation of, 87 primitive irreducible digraph, 455 periodic, 206, 305 primitive irreducible matrix, 445 trace, 236, 240 primitive polynomial, 228 Shannon Coding Theorem, 139 enumeration of, 229 for BSC, 10 product code, 44, 178 with error detection, 132 projective geometry, 47, 361 for q-ary erasure channel, 135 Prouhet–Tarry problem, 329 for q-ary symmetric channel, 117 Shannon Converse Coding Theorem, 139 q-ary erasure channel, 15, 25, 126, 134, 391, for BSC, 10 514 for q-ary erasure channel, 134 q-ary symmetric channel, 4, 19, 24, 32, 110, for q-ary symmetric channel, 113 113, 117, 125, 133, 145, 378, 390, shift register, 198, 217, 237 394, 424, 467, 490 signal-to-noise ratio (SNR), 18 quadratic polynomial, 78, 88 simplex code, 41, 120 Quadratic Reciprocity Law, 411, 449 weight distribution of, 99 quadratic residue, 77, 239, 410, 447 Singleton bound, 94, 136, 332, 334 Euler’s criterion for, 80, 448 asymptotic, 105, 421, 451 Gauss’ criterion for, 448 cover-metric, 354, 362 rank-metric, 353, 361 random coding, 137 size (of codes), 5 Gallager’s error exponent of, 143 spectral-null code, 329 rate, 5 sphere, 12 of convolutional codes, 477 Lee, 317 of trellis codes, 462 volume, 95 recurrence order, 197, 214 sphere-covering bound, 123 redundancy, 27 sphere-packing bound, 95, 122, 136 Reed–Muller code, 41, 48, 177, 260, 265 asymptotic, 107 first-order, 41, 46, 120, 139, 327, 385, Lee-metric, 318, 330 408 splitting field, 65 Reed–Solomon code, see RS code stack algorithm, 520 Reiger bound, 122 standard array decoding, 33 remaindering circuit, 73, 153 state (in digraphs), 453 repetition code, 28 subfield sub-code, 174 ring, 522 subgroup, 521 Roos bound, 265 normal, 522 root, 59 subring, 522 of BCH codes, 163, 250 Sudan’s algorithm, 274, 296 of bivariate polynomials, 268 Sugiyama et al. (SKHN) algorithm, 215 computation, 90, 284, 296 super-multiplicative sequence, 331 of cyclic codes, 248 super-regular matrix, 335 multiplicity, 60, 76 Sylvester-type Hadamard matrix, 45, 49, of RS codes, 152, 163, 250 406, 436, 447 RS code synchronization error, 327 conventional, 151, 243 syndrome, 34

566 Index

polynomial, 185 Viterbi’s algorithm, 468, 519 volume tag (of edges), 459 of Hamming spheres, 95 term rank (of arrays), 362 of Lee spheres, 317 trace of elements in a ﬁnite ﬁeld, 83 weight, 6 of matrices, 331, 514 cover, 354, 362 sequence, 236, 240 distribution, see weight distribution transfer matrix (of bipartite graphs), 399 enumerator, 99, 486 trellis code, 460 Lee, 299, 448 decoding of, 466 weight distribution, 99 encoding of, 464 of cycles in LFSMs, 487 free distance of, 463 of Hamming codes, 101, 121 triangle inequality, 6 of MDS codes, 104 turbo code, 519 of simplex codes, 99 Weil’s Theorem, 179 union bound, 137 Welch–Berlekamp unit element (in a ring), 190 algorithm, 215, 217 unity element, 521 equations, 211, 215, 271, 291, 295 Wyner–Ash code, 512 variable node, 449 vertex (in graphs), 396 Zyablov bound, 373, 392, 413, 422, 438, 440