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

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

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

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 Bézout’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

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 code residue, 496 Hermitian self-orthogonal, 7 reverse circulant construction, 377 hexacode, see hexacode reverse circulant generator matrix, 377 hold a design, 293 self-complementary, 435 homogeneous, 271 self-dual, see self-dual code hull of, 275 self-orthogonal, 6, 310, 340, 360, 363, 469 indecomposable, 368 shortened, 16 information set, 4 simplex, see simplex code inner, 201 singly-even, 12 interleaved, 203 strength of, 435 isodual, 378 subﬁeld subcode, see subﬁeld subcode Kerdock, see Kerdock code sum, 135 lattice from, 427, 503 Tanner graph of, 593 lexicode, 97 ternary, 3 linear, 2, 4 t-error-correcting, 41 low density parity check, see low density parity tetracode, see tetracode check code torsion, 496 maximum distance separable, see maximum trace of, see trace code distance separable code turbo, see turbo code minimum distance, 8 weight distribution of, see weight distribution minimum support weight, see generalized weight enumerator of, see weight enumerator Hamming weight weight hierarchy of, 283, 284 minimum weight, 8 weight spectrum of, see weight distribution monomial automorphism group, 26 Z4-linear, see Z4-linear code monomially equivalent, 24, 281 codeword, 3 nearly perfect, binary, 69 codeword associated with a path, 555 nonlinear, 53 codeword error, 569 Nordstrom–Robinson, see Nordstrom–Robinson coding gain, 578 code commutative ring with unity, 101

636 Subject index

compact disc (CD) recorder, 203 coset, 41 decoding, 207 complete weight distribution of, 45 encoding, 204 cyclotomic, 114, 122 complement of a vector, 333 leader, see coset leader complete coset weight distribution, 45 of nonlinear code, 433 complete decoding, 40 weight distribution of, 265 component, 370 weight of, 41 conjugate, 7 coset leader, 42, 51, 434 conjugate elements, 114 ancestor, 460 conjugation, 7 child, 460 constant weight code, 60, 282 descendant, 460 constraint length, 561 orphan, 460 Construction A, 427 parent, 460 Construction A4, 503 cover, 459 Consultative Committee for Space Data Systems, covering radius ρ(C), 50, 51, 57, 265, 432 612 Adams Bound, 455 convolutional code, 546, 612 amalgamated direct sum (ADS), 452 basic generator matrix, 559 BCH code, 440, 441, 443, 444 canonical generator matrix, 558 Calderbank Bound, 457 catastrophic generator matrix, 569 Delsarte Bound, 440 constraint length, 561 Hamming code, 448 degree of, 558 Hou Bound, 458 delay, 548 length function, 447 encoder, 547 Norse Bounds, 435 recursive systematic, 607 Redundancy Bound, 433 standard form of, 607 Reed–Muller code, 437, 438 state of, 551–553 self-dual code, 444 systematic, 607 simplex code, 439 external degree, 558 Sphere Covering Bound, 434 Forney indices, 561 subcode, 454 free distance, 563 Supercode Lemma, 434 General Viterbi Algorithm, 584, 586 Cross-Interleaved Reed–Solomon Code (CIRC), generator matrix, 547 204 systematic, 607 crossover probability, 39, 583 internal degree, 559 cyclic code, 121 memory, 546, 548, 553, 561 BCH, see BCH code overall constraint length, 561 BCH Bound, 151 polynomial generator matrix, 547 check polynomial, 146 predictable degree property, 559 complement, 145, 211 quick-look-in, 612 defining set, 145 rate of, 546 generating idempotent, 145 reduced generator matrix, 559 generator polynomial, 145 Soft Decision Viterbi Decoding Algorithm, 580, cyclic shift, 121 581, 612 defining set, 142, 144 state diagram, 551, 552 divisor of, 157 trellis diagram, 554 duadic, see duadic code codeword associated with a path, 555 dual, 127, 146 message associated with a path, 555 defining set, 146 survivor path, 555 generating idempotent, 146 truncated, 555 generator polynomial, 146 weight of a path, 555 nonzeros, 146 weight of an edge, 555 encoding, 128 Two-Way a Posteriori Probability (APP) Decoding equivalence, 141 Algorithm, 587, 592 class, 144, 233, 365 Viterbi Decoding Algorithm, 551, 556 extended, 162, 229 coordinate functional, 390 defining set, 164

637 Subject index

generating idempotent, 132, 135 Permutation Decoding Algorithm, 402, 403 computing from generator polynomial, 133 Peterson–Gorenstein–Zierler Algorithm, 179, 182 generator matrix from, 134 soft decision, 573 generator matrix, 125 Soft Decision Viterbi Algorithm, 580, 581, 612 generator polynomial, 126, 135, 144 Sudan–Guruswami Algorithm, 195, 196 computing from generating idempotent, 133 Sugiyama Algorithm, 190, 191 Hartmann–Tzeng Bound, 153 Sum-Product Algorithm, 602 Hermitian dual, 149 Syndrome Decoding Algorithm, 42, 43 defining set, 149 Turbo Algorithm, 610 generating idempotent, 149 turbo code, 607 Hermitian self-orthogonal, 149 Two-Way a Posteriori Probability (APP), 587, 592 irreducible, 150 Viterbi Algorithm, 551, 556 Meggitt Decoding Algorithm, 158–160 degree of a convolutional code, 558 minimum weight, 151, 153 degree of a point, 528 nonzeros, 142 degree of a row, 553, 558 over Z4, see cyclic code over Z4 degree of a vector, 559 parity check matrix, 127, 143 delay, 548 permutation automorphism group of, 139 delay element, 129 primitive, 162 Delsarte Bound, 440 primitive idempotent, 136 Delsarte’s Theorem, 119 quadratic residue, see quadratic residue code demodulation, 207 self-orthogonal, 147 demodulator, 574 defining set, 147 descendant, 460 generator polynomial, 148 design, 291 subfield subcode, 128 Assmus–Mattson Theorem, 303 van Lint–Wilson Bounding Technique, 154 automorphism group of, 292 zeros, 142 automorphism of, 292 cyclic code over Z4, 475 balanced incomplete block design, 294 dual, 485 block of, 291 generating idempotents, 486 code from, 308 generator polynomials, 485 complementary, 298, 302 generating idempotents, 485–487 derived, 298, 302 generator polynomials, 482 equivalent, 292 quadratic residue, 490, 492 extended Pascal triangle, 297 extended, 492 Hadamard, 334 Leech lattice from, 505 held by a code, 293 self-dual, 502 held by binary Golay code, 299, 300, 305, 306, 401 cyclotomic coset, 114, 122 octad, 300 q-, 114 sextet, 300 tetrad, 299 decoding, 39 held by duadic code, 321 Berlekamp–Massey Algorithm, 186, 188 held by extremal code, 349 compact disc, 207 held by Hamming code, 292, 293, 297, 299, 306 complete, 40 held by Pless symmetry code, 421 erasure, 44 held by Reed–Muller code, 306 Gallager Hard Decision Algorithm, 599 held by ternary Golay code, 305 General Viterbi Algorithm, 584, 586 incidence matrix, 291 λ j hard decision, 573 intersection numbers i , 295 hexacode decoding of Golay code, 407 Pascal triangle, 296 iterative, 593, 598, 599, 602, 607, 610 point of, 291 list-decoding, 195 projective plane, see projective plane maximum a posteriori probability (MAP), 39 quasi-symmetric, 314 maximum likelihood (ML), 40, 580 replication number, 294 Meggitt Algorithm, 158–160 residual, 299, 302 Message Passing Algorithm, 595 self-complementary, 298 nearest neighbor, 40 simple, 291

638 Subject index

design (cont.) eight-to-fourteen modulation (EFM), 206 Steiner system, 291 Elias Bound, 74 symmetric, 291 asymptotic, 93 automorphism of, 321, 322 elliptic curve, 528–533, 538 Bruck–Ryser–Chowla Theorem, 319 encoder, 1, 547 code from, 309, 310 standard form of, 607 order of, 291 systematic, 37, 607 difference set, 322 encoding, 37 cyclic, 322 compact disc, 204 development of, 323 cyclic code, 128 multiplier, 327 energy of a signal, 573 normalized block, 328 energy per bit, 577 symmetric design, 323 entropy, 90 direct sum, 18 equivalence class, 144, 233, 365, 518 discrepancy, 187 equivalent codes, 25, 384 discrete memoryless channel (DMC), 39, monomially, 24, 281, 469 576 permutation, 20, 468 8-ary output, 576 equivalent designs, 292 distance, 7, 563 equivalent Hadamard matrices, 331 minimum, 8 equivalent lattices, 424 relative, 89, 541 erasure, 44 distance distribution, 75, 472 error, 44 distance invariant, 472 burst, 200 divisible code, 11, 338 error evaluator polynomial, 190 Gleason–Pierce–Ward Theorem, 339, 389 error function, 575 Division Algorithm, 102 error location number, 180 divisor, 11, 86, 157, 338 error locator polynomial, 181, 196 divisor of a rational function, 533 error magnitude, 180 divisor on a curve, 531 error vector, 1, 40, 179 degree of, 531 Euclidean Algorithm, 102 effective, 531 Euclidean weight, 470 intersection, 531 Euler φ-function, 105, 141, 217 support, 531 Euler totient, 105, 217 DMC, 39 evaluation map, 175, 534 dodecacode, 388 even code, 11 doubly-even code, 12, 361 even-like code, 12, 210 number of, 361 even-like vector, 12, 209 doubly-even vector, 275, 277 extend, 14 duadic code, 209 extended code, 14 codeword weights, 229 Golay G12,33 dual, 223 Golay G24,32 even-like, 210 Hamming, 15, 29 existence, 220, 222 external degree, 558 extended, 226 extremal code, 346, 386 generating idempotent, 210, 233 extrinsic term, 609 Hermitian dual, 223, 224 Hermitian self-orthogonal, 223 Fermat curve, 526, 527, 529, 539 minimum weight, 231, 233, 234 ﬁeld, 2, 100 odd-like, 210 adjoining a root, 108 quadratic residue, see quadratic residue code automorphism, 111 self-orthogonal, 222, 224, 321 binary, 3 splitting, 210 characteristic of, 100 splitting of n, 212 conjugate elements, 114 Square Root Bound, 230 extension, 111 dual code, 5, 469 ﬁnite, 100 Hermitian, 7 order of, 100

639 Subject index

Frobenius automorphism, 112 extended narrow sense RS, 176 Galois group, 112 generator matrix, 177 normal basis, 441 parity check matrix, 177 prime subfield, 100 Sudan–Guruswami Decoding Algorithm, 195, 196 primitive element, 104 Generalized Singleton Bound, 286 quaternary, 3 generator matrix, 4 splitting, 122 of code, 4 subfield, 110 of lattice, 423 ternary, 3 standard form of, 4, 21, 469 trace function Trt , 119 systematic, 607 field of rational functions, 533 genus of a curve, 532 finite weight matrix, 570 Gilbert Bound, 86 First MRRW Bound, 94 asymptotic, 94, 541 fixed element, 112 Giotto, 613 flip-flop, 129 Gleason’s Theorem, 341, 385 flow along a path, 584 Gleason–Pierce–Ward Theorem, 339, 389 flow between vertices, 584 Gleason–Prange Theorem, 249 form H, O, or A, 277, 393 glue element, 370 formally self-dual code, 307, 338 glue vector, 370 Balance Principle, 379 Golay codes, binary G23, G24, 32, 49, 397, 613 extremal, 346 as duadic code, 211 isodual, 378 as lexicode, 99 minimum distance, 344, 345 as quadratic residue code, 240, 401 Forney indices, 561 automorphism group, 251 frame, 204 complete coset weight distribution of, 306 free distance, 563 covering radius, 402 free module, 467 decoding with the hexacode, 407 basis, 467 design held by, 299, 300, 305, 306, 401 Frobenius automorphism, 112, 528 extended, 32 generating idempotent of, 401 Galileo, 613 generator polynomial of, 401 Gallager Hard Decision Decoding Algorithm, 599 Leech lattice from G24, 429 Galois field, see field octad, 300, 429 Galois group, 112, 506 odd Golay code, 358 Galois ring, 505 PD-set of, 404 2-adic representation, 506 perfect, 48, 49 Frobenius automorphism, 506 permutation decoding, 404 Galois group, 506 sextet, 300, 429 primitive element, 506 tetrad, 299, 429 primitive polynomial, 507 uniqueness of, 49, 401 General Viterbi Algorithm, 584, 586 weight enumerator of, 261, 272, 302 Generalized Griesmer Bound, 287 Golay codes, ternary G11, G12, 33, 49, 413 generalized Hamming weight, 289 as duadic code, 213 generalized Hamming weight dr (C), 283 as quadratic residue code, 243, 418 dual code, 284 automorphism group, 419 extended binary Golay code, 285 covering radius, 419 Generalized Griesmer Bound, 289 design held by, 305 simplex code, 283, 290 extended, 33 generalized Reed–Muller (GRM) code Rq (r, m), 524 generating idempotent of, 134, 418 order of, 524 generator polynomial of, 134, 418 generalized Reed–Solomon (GRS) code GRSk (γ, v), perfect, 48, 49 176, 520 uniqueness of, 49, 418 as algebraic geometry code, 537 weight enumerator of, 260, 272 as MDS code, 176, 178 Goppa code (L, G), 521 dual, 176 meets Asymptotic Gilbert–Varshamov Bound, 542 extended, 177 parity check matrix, 522

640 Subject index

Graeffe’s method, 479 hard decision decoding, 573 Gram matrix, 423, 504 Hartmann–Tzeng Bound, 153 graph, 456 Hensel’s Lemma, 477 chromatic number, 456 Hermitian curve, 527, 529, 538, 539 coloring of, 456 Hermitian dual, 7 edge, 456 Hermitian inner product, 7 Kneser, 456 Hermitian self-dual code, 7, 26, 228, 234, 245, 338, Tanner, 593 344 vertex, 456 Classiﬁcation Algorithm, 367 Gray map, 472 design from, 349 Green machine, 611 Gleason’s Theorem, 341 Griesmer Bound, 81 mass formula, 366 generalized, 287 minimum distance, 344, 345 generalized Hamming weight, 289 number of, 362 group, 22 Type IV, 339, 362 afﬁne, 165, 251, 366, 368 extremal, 346 automorphism, 22, 26, 139, 292, 384, hexacode G6, 7, 383, 405 469 as duadic code, 214 Galois, 112, 506 as extended Hamming code, 406 Mathieu, 251, 402, 419 as quadratic residue code, 240 of units, 217 automorphism group of, 27, 406 order of an element, 105 covering radius, 406 projective general linear, 422 to decode the extended binary Golay code, 407 projective special linear, 22, 249, 251, uniqueness of, 405 402 Hilbert entropy function Hq (x), 90, 541 symmetric, 22 homogeneous coordinates, 518 transitive, 23 homogenization, 519 GRS, 176 Hou Bound, 458 hull of a code, 275 H-plane, 275 Huygens, 614 Hadamard matrix, 331, 612 design from, 334 ideal, 106 equivalent, 331 minimal, 136 normalized, 331 primary, 476 order of, 331 principal, 106, 125 Reed–Muller code, 333 idempotent, 132 Hamming Bound, 48 generating, 132, 485, 487 asymptotic, 92 primitive, 136 Hamming code Hr or Hq,r ,5,29 incidence matrix, 291 as BCH code, 169 independent sequence, 154 as duadic code, 211 information rate, 47, 88 as lexicode, 99 information set, 4, 13, 71 automorphism group of H3,22 information transmission rate, 577 automorphism group of H3, 251 inner distribution, 75 complete coset weight distribution of H3, 45, integral domain, 101 266 principal ideal domain, 106 design held by, 292, 293, 297, 299, 306 interleave to depth t, 203 dual of, 30 interleaver, 604 generating idempotent of H3, 133, 142 internal degree, 559 generator polynomial of H3, 133, 142 intersection multiplicity, 529 Gosset lattice from H3, 428 iterative decoding, 593, 598, 599, 602, 607, Syndrome Decoding Algorithm, 43 610 weight distribution of, 261 word error rate of H3,46 Jet Propulsion Laboratory, 612 Hamming distance distribution, see distance Johnson Bound, 65, 74 distribution JPL, 612

641 Subject index

Kerdock code K(r + 1), 509 MacWilliams equations, 252, 253, 257, 384 weight distribution, 513 MAP, 39 key equation, 191 Mariner, 611 Klein quartic, 527, 529, 540 mass formula, 365, 386, 498, 499 n,q Krawtchouck polynomial Kk (x), 75, 256 Massey–Sain Theorem, 570 Mathieu group, 251, 402, 419 Lagrange Interpolation Formula, 537 matrix, 4 land, 204 bordered circulant, 31, 376 lattice, 423, 503 bordered reverse circulant, 377 basis, 423 circulant, 376 Construction A, 427 ﬁnite weight, 570 Construction A4, 503 generator, 4, 547 density, 428 standard form of, 4, 21 determinant, 423 Gram, 423, 504 discriminant, 423 Hadamard, see Hadamard matrix dual, 424 incidence, 291 equivalent, 424 minor of, 559 from code, 427, 503 monomial, 24 fundamental parallelotope, 423 diagonal part of, 24 generator matrix, 423, 427, 504 permutation part of, 24 Gosset, 428, 505 parity check, 4 Gram matrix, 423, 504 permutation, 20 integral, 423 reverse circulant, 32, 377 even, 425 right inverse, 38 odd, 425 unimodular, 559 self-dual, 424 Vandermonde, 151 Type I, 425, 427, 504 maximum a posteriori probability (MAP) decoding, Type II, 425, 427, 504 39 unimodular, 424 maximum distance separable (MDS) code, 71 kissing number, 426 bound, 264 Leech, 429, 505 extended GRS, 177 norm, 425 generalized Reed–Solomon, 176 packing, 426 Generalized Singleton Bound, 286 planar hexagonal, 423 MDS Conjecture, 265 point of, 423 Reed–Solomon, 174 theta series, 426 trivial, 72, 264 Law of Quadratic Reciprocity, 219 weight distribution of, 262 LDPC, 598 maximum likelihood (ML) decoding, 40, 580 Lee weight, 470 McEliece’s Theorem, 157 Legendre symbol, 219 MDS, 71 lexicode, 97 MDS Conjecture, 265 binary Golay, 99 meet, 53 binary Hamming, 99 Meggitt Decoding Algorithm, 158–160 line, 230 memory, 546, 548, 553, 561 linear code, 2 merge bits, 206 linear feedback shift-register, 129, 190 message, 1 linear feedforward shift-register, 549 message associated with a path, 555 Linear Programming Bound, 78 message error, 569 asymptotic, 94 message passing, 594 linear shift-register, 129 Message Passing Decoding Algorithm, 595 list-decoding algorithm, 195 minimum distance, 8 low density parity check (LDPC) code, 598 minimum support weight dr (C), see generalized Gallager Hard Decision Decoding Algorithm, Hamming weight 599 minimum weight, 8 Sum-Product Decoding Algorithm, 602 even-like, 15 Lucas’ Theorem, 166 odd-like, 15

642 Subject index

minor, 559 parity check matrix, 4 ML, 40 partial derivative, 526 modulator, 573 PCCC, 604 monomial automorphism group, 26, 469 PD-set, 403 monomial matrix, 24 of binary Golay code, 404 diagonal part of, 24 perfect code, 48, 49 permutation part of, 24 Golay, 48 monomially equivalent codes, 24, 281, 469 Hamming, 48, 49 MRRW, 93 trivial, 49 multiplicity at a point, 529 permutation automorphism group, 22, 139, 469 multiplicity of a root, 195 Permutation Decoding Algorithm, 402, 403 multiplier, 138, 210, 486 binary Golay code, 404 of difference set, 327 PD-set, 403 splitting of n by, 212 permutation equivalent codes, 20, 468 permutation matrix, 20 nearest neighbor decoding, 40 permuter, 604, 607 nearly perfect binary code, 69 Peterson–Gorenstein–Zierler Decoding Algorithm, punctured Preparata code, 70 179, 182 Newton identities, 187 Pioneer, 612 Newton radius, 465 pit, 204 noise, 1, 40, 574 Pless power moments, 256, 259 additive white Gaussian, 575 Pless symmetry code, 420 nonlinear code, 53 automorphism group, 421, 422 nonsingular curve, 526 design held by, 421 nonsingular point, 526 extremal, 421 Nordstrom–Robinson code N 16,68 Plotkin Bound, 58 from octacode, 475 asymptotic, 89 optimal, 69 Plücker’s Formula, 532 uniqueness of, 69 point, 29, 195, 230, 291, 423, 517, 518 weight distribution of, 69 affine, 518 norm of a code, 451 at infinity, 518 norm of a vector, 425 degree of, 528 normal code, 452 multiplicity at, 529 normalized set, 328 nonsingular, 526 Norse Bounds, 435 rational, 528 Nyquist rate, 578 simple, 526 singular, 526 octad, 300, 429 polynomial, 101 odd-like code, 12, 210 basic irreducible, 476 odd-like vector, 12, 209 check, 146, 510 optimal code, 53 coefficient of, 101 Nordstrom–Robinson, 69 coprime, 477 order ordn(q), 122 degree of, 101 order of a group element, 105 derivative, 122 orphan, 460 divides, 102 orthogonal code, 5 Division Algorithm, 102 orthogonal sum, 276 divisor of, 102 orthogonal vectors x⊥y, 275 error evaluator, 190 oval, 318 error locator, 181, 196 overall constraint length, 561 Euclidean Algorithm, 102 overall parity check, 15 factor of, 102 generator, 126 packing radius, 41 generator matrix, 547 parallel concatenated convolutional code (PCCC), 604 Gleason, 342 parallel concatenated encoder, 604 Goppa, 522 parent, 358, 460 greatest common divisor, 102

643 Subject index

homogeneous, 519 point at infinity, 518 homogenization of, 519 projective special linear group, 22, 249, 251, 402 irreducible, 106, 476 puncture, 13 Krawtchouck, 75, 256 punctured code, 13 leading coefficient of, 102 hexacode, 214, 240 minimal, 112 Preparata, 70 monic, 102, 476 Reed–Muller, 225 primary, 476 primitive, 108, 507 QR, 237 reciprocal, 116, 145, 483 quadratic non-residue, 237 regular, 478 quadratic residue, 237, 323, 335 relatively prime, 102, 477 quadratic residue (QR) code, 237 reversible, 145 automorphism group, 248, 251 syndrome, 159 binary Golay code, 240, 246, 250 term of, 101 existence, 237, 239, 241, 243, 244 degree of, 101 extended, 245–248 weight enumerator, see weight generating idempotent, 238, 239, enumerator 241–244 weighted degree of, 195 Gleason–Prange Theorem, 249 Prange’s Theorem, 271 Hermitian self-orthogonal, 241 predictable degree property, 559 hexacode, 246, 250 Preparata code P(r + 1), 515 minimum weight, 249 as a nearly perfect code, 70 over Z4, 490, 492 weight distribution, 516 punctured hexacode, 240 primary ideal, 476 self-orthogonal, 241, 244 prime subfield, 100 ternary Golay code, 243, 246, 250 primitive element, 104 quantization, 575 primitive root of unity, 105, 122 binary, 575 principal ideal, 106, 125 quantum error-correction, 383 principal ideal domain, 106 quasi-perfect code, 50 probability, 39 quaternary field, 3 a posteriori, 587, 608 quick-look-in convolutional code, 612 a priori, 608 crossover, 39, 583 radius, 40 maximum likelihood, 40 covering, see covering radius projective closure, 526 Newton, 465 projective general linear group, 422 packing, 41 projective geometry, 29, 319 rate, 47, 88, 546 point of, 29, 170 rational point, 528 projective line, 518 receiver, 1 projective plane, 230, 291, 315, 518 recursive systematic convolutional (RSC) encoder, 607 as a symmetric design, 315 reduced generator matrix, 559 code from, 316 reduction homomorphism, 476 cyclic, 230, 321 redundancy, 1, 4 nonexistence, 324, 326 Redundancy Bound, 433 held by duadic code, 231 redundancy set, 4 line of, 230 Reed–Muller (RM) code R(r, m), 33 of order ten, 329 as duadic code, 225 order of, 230, 291 covering radius, 437, 438 oval, 318 design held by, 306 point of, 230 dimension of, 34 projective plane curve, 526 dual of, 34 genus, 532 from Hadamard matrix, 333 projective space, 518 generalized, see generalized Reed–Muller code affine point, 518 minimum weight of, 34 homogeneous coordinates, 518 order of, 34

644 Subject index

Reed–Solomon (RS) code, 173, 520, 613 Type II, 339, 361, 363, 427, 495 as algebraic geometry code, 536 extremal, 346 as MDS code, 174 Type III, 339, 362 burst error-correcting, 202 extremal, 346 cross-interleaved (CIRC), 204 Z4-, 469, 495 encoding, 175 self-orthogonal code, 6, 316, 360, 363, 368, extended narrow sense, 176 469 generalized, see generalized Reed–Solomon code Hermitian, 7 Peterson–Gorenstein–Zierler Decoding Algorithm, number of, 360, 363 179, 182 semiring, 584 Sudan–Guruswami Decoding Algorithm, 195, 196 sextet, 300, 429 Sugiyama Decoding Algorithm, 190, 191 shadow, 353, 355, 356 regular curve, 526 Shannon, Claude, 1 relative distance, 89 Shannon limit, 577, 603 relative trace function TRr , 508 Shannon’s Theorem, 2, 46, 47, 577 replicated code, 390 shorten, 16 residual code Res(C, c), 80 shortened code, 16 residue class ring, 107, 121 MDS, 202 residue of a function, 524 Reed–Solomon, 202, 204 Restricted Johnson Bound, 61 signal-to-noise ratio, 577 reverse circulant matrix, 32, 377 signaling power, 577 Riemann–Roch Theorem, 535 simple point, 526 right inverse, 38 simplex code, 30, 36, 82, 282 ring, 101, 475 covering radius, 439 commutative, 101 generalized Hamming weight, 283, Galois, see Galois ring 290 ideal of, 106 meet the Griesmer Bound, 82 integral domain, 101 weight distribution of, 30, 82 residue class, 107, 121 Singleton Bound, 71 with unity, 101, 132 asymptotic, 89 RM, 34 generalized, 286 root of unity, 105 singly-even code, 12 RS, 173 singly-even vector, 275, 277 RSC, 607 singular point, 526 smooth curve, 526 Second MRRW Bound, 94 soft decision decoding, 573 self-complementary code, 435 Soft Decision Viterbi Decoding Algorithm, 580, 581, self-dual code, 6, 26, 227, 229, 230, 233, 234, 612 245–248, 317, 338, 340, 344, 359, 364, source, 1 469 sphere, 40 Balance Principle, 351 Sphere Covering Bound, 434 bound, 344, 346, 495 Sphere Packing Bound, 48, 59, 74 child, 358, 375 splitting, 210 Classification Algorithm, 366 splitting field, 122 covering radius, 444 splitting of n, 212 design from, 349 Square Root Bound, 230 Gleason polynomials, 342 state diagram, 551, 552 Gleason’s Theorem, 341 state of an encoder, 551–553 Hermitian, see Hermitian self-dual code Stirling numbers S(r,ν), 256 mass formula, 366 Stirling’s Formula, 90 minimum distance, 344–346 strength, 435 number of, 359, 362, 374, 375 subcode, 5 parent, 358 subfield subcode C| , 116 Fq shadow, 353, 355, 356 dual of, 119 Type I, 339, 427, 495 of a cyclic code, 128 extremal, 346 parity check matrix, 117

645 Subject index

Sudan–Guruswami Decoding Algorithm, 195, 196 parallel concatenated convolutional code, 604 Sugiyama Decoding Algorithm, 190, 191 parallel concatenated encoder, 604 Sum-Product Decoding Algorithm, 602 permuter, 604, 607 Supercode Lemma, 434 Turbo Decoding Algorithm, 610 support supp(D) of a code, 283 Turbo Decoding Algorithm, 610 support supp(c) of a vector, 120 two-sided power spectral density, 575 survivor path, 555 Two-Way a Posteriori Probability (APP) Decoding symmetry code, see Pless symmetry code Algorithm, 587, 592 syndrome, 42, 51, 179 Type I code, 339, 385, 495 Syndrome Decoding Algorithm, 42, 43 extremal, 346, 386 systematic encoder, 37, 607 Type II code, 339, 361, 363, 385, 495 systematic generator matrix, 607 extremal, 346, 386 systematic term, 609 Type III code, 339, 362 extremal, 346 Tanner graph, 593 Type IV code, 339, 362 check node, 594 extremal, 346 Message Passing Decoding Algorithm, 595 variable node, 594 (u | u + v) construction, 19 t-design, see design unimodular matrix, 559 tensor product, 332 unique factorization domain, 106 ternary field, 3 unit, 217 t-error-correcting code, 41 unity, 132 tetracode H3,2,6 Unrestricted Johnson Bound, 63 as Hamming code, 30 automorphism group of, 26 van Lint–Wilson Bounding Technique, 154 tetrad, 299, 429 Vandermonde matrix, 151 theorem Varshamov Bound, 88 Assmus–Mattson, 303 asymptotic, 94, 541 Bruck–Ryser–Chowla, 319 vector, 1 Bézout, 531 ρ-covered, 432 Delsarte, 119 complement of, 333 Gleason, 341 cover, 459 Gleason–Pierce–Ward, 339 doubly-even, 275, 277 Gleason–Prange, 249 error, 1, 40, 179 Hensel’s Lemma, 477 even-like, 12, 209 Law of Quadratic Reciprocity, 219 glue, 370 Lucas, 166 odd-like, 12, 209 Massey–Sain, 570 received, 1, 41 McEliece, 157 singly-even, 275, 277 Plücker’s Formula, 532 support of, 120 Prange, 271 syndrome of, 42, 179 Riemann–Roch, 535 trace of, 119 Shannon, 2, 46, 47, 577 Viking, 611 Supercode Lemma, 434 Viterbi Decoding Algorithm, 551, 556 trace code Trt (C), 119 Voyager, 613 dual of, 119 trace function Trt , 119 weight, 8, 470, 562 transitive, 23 coset, 41 trellis diagram, 554 distribution, see weight distribution flow along a path, 584 enumerator, see weight enumerator flow between vertices, 584 Euclidean, 470 truncated trellis, 555 generalized Hamming, see generalized Hamming turbo code, 602 weight decoding, 607 Hamming, 8, 470 encoding, 604 hierarchy, 283, 284 interleaver, 604 Lee, 470

646 Subject index

weight, (cont.) Hamming distance, 470 minimum, 8 Hamming distance distribution, 472 minimum even-like, 15 Hamming weight, 470 minimum odd-like, 15 Hamming weight enumerator, 471 minimum support, see generalized Hamming Kerdock, see Kerdock code weight lattice, see lattice spectrum, see weight distribution Lee distance, 470 weight distribution, 252 Lee weight, 470 MacWilliams equations, 252, 253 Lee weight enumerator, 471 Pless power moments, 256, 259 monomial automorphism group, 469 weight enumerator WC (x)orWC (x, y), 255, 257 monomially equivalent, 469 complete, 273 octacode, 474 Lee, 274 Gosset lattice from, 505 weight hierarchy, 283, 284 Nordstrom–Robinson code from, weight of a path, 555 475 weight of an edge, 555 permutation automorphism group, 469 weight preserving linear transformation, permutation equivalent, 468 279 Preparata, see Preparata code weighted degree, 195 residue, 496 word error rate, 46 self-dual, 469 cyclic, see cyclic code over Z4 Z4-linear code, 467 Euclidean-extremal, 495 complete weight enumerator, 471 generator matrix, 497 cyclic, see cyclic code over Z4 mass formula, 498, 499 distance invariant, 472 number of, 498, 499, 501 dual, 469 Type I, 495, 504 Euclidean distance, 470 Type II, 495, 504 Euclidean weight, 470 self-orthogonal, 469 generator matrix, 468 symmetrized weight enumerator, 471 standard form of, 469 torsion, 496 Gray map, 472 type, 468