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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. Cary Huffman and Vera Pless Index More information 631 Symbol index extdeg G, 558 mC, 368 F2,3 N0, 575 F3,3 N 16,68 F4,3 na (x), 470 k F9, 243 (n, 2 , d), 383 F16, 184 (n, c, r), 598 Fq ,2,100 (n, k, m), 561 Fn , , , q ,3 (n k m d), 563 Fq [x], 101 (n, M, d), 53 F[x1,...,xn], 519 N p, 237 H f , 519 Ns (δ), 195 fn, 373 N(v), 426 f (x) | g(x), 102 ordn(q), 122 G6,7 PAut(C), 22, 469 G11,33 (P, B), 291 G12,32 Pb, 578 G23,32 Perr,46 s G24,31 PG(2, 2 ), 319 g24, 374 PG(r − 1, q), 29 GA1(I), 165 PGL2(q), 422 GA3(2), 368 Pk , 174 Gal(Fq ), 112 P(r + 1), 515 Gal(Fq : Fpr ), 112 prob(E1), 39 G(C), 371 prob(E1 | E2), 39 G0(C), 372 Pr(v), 407 G1(C), 372 PSL2(7), 22, 368 G2(C), 371 PSL2(23), 402 G(C), 472 Qp, 237 gcd( f (x), g(x)), 102 Res(C), 496 GF(q), 2, 100 Res(C, c), 80 r GR(4 ), 505 Resγi f , 524 GRSk (γ, v), 176, 196, 520 Rn, 121, 124, 209 H3,5 Rn, 480 H3,6 R(r, m), 34 Hr ,29 Rq (r, m), 524 H2,r ,29 Rt , 577 H3,2,6 span{x, y}, 275 Hq,r ,29 S(q), 420 Hq (x), 90 S(r,ν), 256 HamC (x, y), 470 Sr (u), 40 i2, 367 S(t, k,v), 291 I (C, t), 203 supp(c), 120 intdeg G, 559 supp(D), 283 Jv, 309 Symn,21 j(x), 209 tq (n, k), 447 K(r + 1), 509 Tor(C), 496 n,q T R Kk (x), 75, 256 ( ), 506 L(D), 534 TRr , 508 Lk−1, 520 Trt , 119 q (m, r), 447 Trt (C), 119 LeeC (x,y), 448 Vq (n, a), 74 M12, 419 WC (x), 255 M23, 402 WC (x, y), 255 M24, 402 wt(x), 8 MAut(C), 26, 469 wtE (x), 470 MAutPr(C), 28 wtH (x), 470 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-13170-4 - Fundamentals of Error-Correcting Codes W. Cary Huffman and Vera Pless Index More information 632 Symbol index wtL (x), 470 x⊥y, 275 X 1 ∩ X 2, 531 x ∩ y,8 X f (F), 526 x · y,6,469 x j (i), 587 Zq ,76 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-13170-4 - Fundamentals of Error-Correcting Codes W. Cary Huffman and Vera Pless Index More information Subject index a priori term, 609 Meggitt Decoding, 158–160 α notation, 608 Message Passing Decoding, 595 A-plane, 275 Permutation Decoding, 402, 403 acceptable coordinate, 451 Peterson–Gorenstein–Zierler Decoding, 179, 182 Adams Bound, 455 Soft Decision Viterbi Decoding, 580, 581, 612 additive code over F4, 383 Sudan–Guruswami Decoding, 195, 196 automorphism group, 384 Sugiyama Decoding, 190, 191 Balance Principle, 387 Sum-Product Decoding, 602 dodecacode, 388 Syndrome Decoding, 42, 43 equivalent, 384 Turbo Decoding, 610 generator matrix, 383 Two-Way a Posteriori Probability (APP) Decoding, hexacode, 383 587, 592 mass formula, 386 Viterbi Decoding, 551, 556 trace dual, 384 amalgamated direct sum (ADS), 452 trace inner product, 383 ancestor, 460 trace self-dual, 384 APP, 587 trace self-orthogonal, 384 Assmus–Mattson Theorem, 303 Type I, 385 asymptotic bound, 88 extremal, 386 Elias, 93 Type II, 385 First MRRW, 94 extremal, 386 Gilbert–Varshamov, 94, 541 Gleason’s Theorem, 385 exceeded by algebraic geometry codes, 544 additive white Gaussian noise (AWGN), 575 met by Goppa codes, 542 adjoining a root, 108 Hamming, 92 affine group, 165, 251, 366, 368 Plotkin, 89 affine plane curve, 526 Second MRRW, 94 affine space, 517 Singleton, 89 affine-invariant code, 162, 165 Tsfasman–Vl˘adut¸–Zink, 544 extended BCH, 172 asymptotically bad code, 173, 541 AG, 535 asymptotically good code, 173, 542 algebraic geometry (AG) code C(X , P, D), 535 automorphism group, 22, 26, 384 dimension, 535 monomial, 26, 469 dual, 541 of a design, 292 exceed Asymptotic Gilbert–Varshamov Bound, 544 permutation, 22, 469 generalized Reed–Solomon code as, 537 transitive, 23, 28, 271, 308 generator matrix, 535 automorphism of a design, 292 minimum distance, 535 automorphism of a field, 111 Reed–Solomon code as, 536 fixed element, 112 algorithm Frobenius, 112 Berlekamp–Massey Decoding, 186, 188 Galois group, 112 Classification, 366 AWGN, 575 Division, 102 Euclidean, 102 Balance Principle, 351, 379, 387 Gallager Hard Decision Decoding, 599 bandwidth, 577 General Viterbi, 584, 586 basic generator matrix, 559 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-13170-4 - Fundamentals of Error-Correcting Codes W. Cary Huffman and Vera Pless Index More information 634 Subject index basis of minimum weight codewords, 83, 85 Plotkin, 58 BCH Bound, 151 Redundancy, 433 BCH code, 168 Singleton, 71 Berlekamp–Massey Decoding Algorithm, 186, 188 Sphere Covering, 434 Bose distance, 171 Sphere Packing, 48, 59, 74 covering radius, 440, 441, 443, 444 Square Root, 230 designed distance, 168 Supercode Lemma, 434 dimension, 170 van Lint–Wilson Bounding Technique, 154 minimum distance, 171 Varshamov, 88 narrow-sense, 168, 521 BPSK, 573 nested, 169 Bruck–Ryser–Chowla Theorem, 319 Peterson–Gorenstein–Zierler Decoding Algorithm, BSC, 39 179, 182 burst, 202 primitive, 168 byte, 202 affine-invariant extension, 172 Reed–Solomon code, see Reed–Solomon code Calderbank Bound, 457 Sugiyama Decoding Algorithm, 190, 191 canonical generator matrix, 558 BER, 578 Cassini, 602, 614 Berlekamp–Massey Decoding Algorithm, 186, 188 catastrophic generator matrix, 569 B´ezout’s Theorem, 531 CCSDS, 612 Big Viterbi Decoder, 613 CD, 203 binary adder, 129 Challenger, 613 binary field, 3 channel, 1, 573 binary phase-shift keying (BPSK), 573 binary symmetric, 39, 583 binary symmetric channel (BSC), 39, 583 capacity of, 1, 47, 577 crossover probability of, 39, 583 discrete memoryless, 39 bit, 202 noisy, 1 bit error rate (BER), 578 statistics, 576, 587 block, 291 channel capacity, 1, 47, 577 bordered circulant matrix, 31, 376 characteristic, 100 bordered reverse circulant matrix, 377 child, 358, 375, 460 bound, 48 CIRC, 204 Aq (n, d), 48 circulant matrix, 376 Aq (n, d,w), 60 Classification Algorithm, 366 Bq (n, d), 48 classification problem, 365 Adams, 455 clock cycle, 129, 549, 573 asymptotic, see asymptotic bound code, 3 BCH, 151 additive, see additive code over F4 Calderbank, 457 affine-invariant, see affine-invariant code Delsarte, 440 algebraic geometry, see algebraic geometry (AG) Elias, 74 code Generalized Griesmer, 287 asymptotically bad, 173, 541 Generalized Singleton, 286 asymptotically good, 173, 542 Gilbert, 86 automorphism group, 26 Griesmer, 81 BCH, see BCH code Hamming, 48 binary, 3 Hartmann–Tzeng, 153 block, 546 Hou, 458 bordered double circulant construction, 376 Johnson, 65, 74 bordered double circulant generator matrix, 376 restricted, 61 bordered reverse circulant construction, 377 unrestricted, 63 bordered reverse circulant generator matrix, 377 Linear Programming, 78 burst error-correcting, 202 meet, 53 complement of, 145 MRRW, 94 component, 370 Norse, 435 concatenated, 201 on maximum distance separable code, 264 constant weight, 60, 282 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-13170-4 - Fundamentals of Error-Correcting Codes W. Cary Huffman and Vera Pless Index More information 635 Subject index convolutional, see convolutional code normal, 452 covering radius of, see covering radius odd-like, 12, 210 cyclic, see cyclic code optimal, 53 decomposable, 368 orthogonal, 5 direct sum, 18 orthogonal sum, 276 weight enumerator of, 255 outer, 201 divisible, 11 packing radius of, 41 divisor of, 11, 86, 157 parallel concatenated convolutional, 604 double circulant construction, 376 parity check matrix, 4 double circulant generator matrix, 132, 376 perfect, 48, 49 doubly-even, 12, 150, 361 permutation automorphism group, 22 duadic, see duadic code permutation equivalent, 20 dual, 5, 469 Pless symmetry, see Pless symmetry code equivalent, 25 Preparata, see Preparata code even, 11 punctured, 13 even-like, 12, 210 quadratic residue, see quadratic residue code extended, 14 quasi-cyclic, 131 extremal, 346 quasi-perfect, 50 formally self-dual, see formally self-dual code quaternary, 3 generalized Hamming weight, see generalized quick-look-in, 612 Hamming weight rate of, 47 generator matrix, 4 redundancy set, 4 standard form of, 4, 21 Reed–Muller, see Reed–Muller code Golay, binary, see Golay codes, binary Reed–Solomon, see Reed–Solomon code Golay, ternary, see Golay codes, ternary repetition, 4 Goppa, see Goppa code replicated, 390 Hamming, see Hamming code residual, 80 Hermitian self-dual, see Hermitian self-dual
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