Coding and Information TheoryThis book is an introduction to information and coding theory at the graduate or advanced undergraduate level. It assumes a basic knowledge of probability and modern algebra, but is otherwise self- contained. The intent is to describe as clearly as possible the fundamental issues involved in these subjects, rather than covering all aspects in an encyclopedic fashion. The first quarter of the book is devoted to information theory, including a proof of Shannon's famous Noisy Coding Theorem. The remainder of the book is devoted to coding theory and is independent of the information theory portion of the book. After a brief discussion of general families of codes, the author discusses linear codes (including the Hamming, Golary, the Reed-Muller codes), finite fields, and cyclic codes (including the BCH, Reed-Solomon, Justesen, Goppa, and Quadratic Residue codes). An appendix reviews relevant topics from modern algebra. |
Contents
IV | 11 |
V | 12 |
VI | 16 |
VII | 18 |
VIII | 22 |
IX | 23 |
XII | 26 |
XIII | 28 |
CXLII | 260 |
CXLIII | 262 |
CXLV | 263 |
CXLVII | 267 |
CXLIX | 268 |
CL | 269 |
CLI | 270 |
CLII | 272 |
XIV | 30 |
XVI | 33 |
XVII | 39 |
XVIII | 40 |
XIX | 41 |
XXI | 43 |
XXII | 44 |
XXIII | 47 |
XXIV | 52 |
XXVI | 54 |
XXVII | 56 |
XXVIII | 59 |
XXIX | 62 |
XXX | 64 |
XXXI | 69 |
XXXII | 72 |
XXXIII | 76 |
XXXIV | 81 |
XXXV | 83 |
XXXVI | 84 |
XXXVII | 89 |
XXXVIII | 91 |
XXXIX | 92 |
XL | 93 |
XLI | 95 |
XLII | 96 |
XLIII | 98 |
XLIV | 101 |
XLV | 105 |
XLVI | 106 |
XLVII | 107 |
XLVIII | 111 |
XLIX | 117 |
L | 119 |
LI | 122 |
LIII | 123 |
LV | 126 |
LVI | 129 |
LVII | 131 |
LVIII | 132 |
LIX | 134 |
LX | 136 |
LXI | 138 |
LXII | 143 |
LXIII | 144 |
LXIV | 145 |
LXVI | 149 |
LXVII | 150 |
LXX | 151 |
LXXII | 152 |
LXXIV | 153 |
LXXV | 154 |
LXXVIII | 155 |
LXXX | 156 |
LXXXII | 157 |
LXXXIII | 158 |
LXXXV | 159 |
LXXXVI | 163 |
LXXXVIII | 165 |
LXXXIX | 167 |
XC | 170 |
XCII | 171 |
XCIV | 172 |
XCV | 173 |
XCVII | 174 |
XCIX | 175 |
C | 176 |
CI | 178 |
CII | 181 |
CIII | 183 |
CIV | 188 |
CV | 197 |
CVI | 199 |
CVII | 202 |
CVIII | 205 |
CIX | 206 |
CXI | 208 |
CXII | 210 |
CXIII | 212 |
CXIV | 216 |
CXVI | 218 |
CXVII | 219 |
CXVIII | 220 |
CXIX | 222 |
CXX | 223 |
CXXI | 225 |
CXXII | 226 |
CXXIII | 228 |
CXXIV | 230 |
CXXV | 235 |
CXXVIII | 237 |
CXXIX | 239 |
CXXX | 240 |
CXXXI | 245 |
CXXXII | 246 |
CXXXIII | 250 |
CXXXIV | 251 |
CXXXV | 253 |
CXXXVI | 254 |
CXXXVII | 256 |
CXL | 258 |
CLIII | 274 |
CLIV | 275 |
CLV | 276 |
CLVI | 278 |
CLVII | 285 |
CLVIII | 286 |
CLIX | 287 |
CLX | 288 |
CLXI | 289 |
CLXII | 296 |
CLXIII | 297 |
CLXIV | 299 |
CLXV | 300 |
CLXVI | 301 |
CLXVII | 302 |
CLXVIII | 304 |
CLXIX | 308 |
CLXXI | 309 |
CLXXII | 310 |
CLXXIII | 312 |
CLXXIV | 314 |
CLXXV | 315 |
CLXXVI | 320 |
CLXXVII | 321 |
CLXXVIII | 325 |
CLXXIX | 327 |
CLXXX | 328 |
CLXXXI | 331 |
CLXXXII | 333 |
CLXXXIII | 335 |
CLXXXIV | 336 |
CLXXXV | 342 |
CLXXXVII | 344 |
CXC | 345 |
CXCI | 347 |
CXCII | 349 |
CXCIV | 353 |
CXCV | 354 |
CXCVI | 356 |
CXCVII | 358 |
CXCVIII | 360 |
CXCIX | 362 |
CCI | 363 |
CCV | 365 |
CCVI | 369 |
CCVIII | 370 |
CCIX | 371 |
CCX | 372 |
CCXI | 374 |
CCXII | 375 |
CCXIII | 376 |
CCXIV | 377 |
CCXV | 379 |
CCXVII | 380 |
CCXVIII | 381 |
CCXIX | 382 |
CCXX | 386 |
CCXXII | 389 |
CCXXIII | 390 |
CCXXIV | 393 |
CCXXV | 395 |
CCXXVI | 396 |
CCXXVII | 398 |
CCXXVIII | 401 |
CCXXIX | 407 |
CCXXXI | 409 |
CCXXXII | 412 |
CCXXXIV | 414 |
CCXXXV | 417 |
CCXXXVI | 421 |
CCXXXVII | 422 |
CCXXXVIII | 423 |
CCXXXIX | 425 |
CCXLI | 426 |
CCXLIII | 427 |
CCXLIV | 428 |
CCXLV | 429 |
CCXLVII | 430 |
CCLI | 431 |
CCLIII | 432 |
CCLIV | 433 |
CCLVI | 435 |
CCLVIII | 436 |
CCLIX | 440 |
CCLX | 441 |
CCLXI | 442 |
CCLXII | 443 |
CCLXIII | 445 |
CCLXIV | 446 |
CCLXV | 449 |
CCLXVII | 452 |
CCLXVIII | 456 |
CCLXIX | 459 |
CCLXX | 463 |
CCLXXI | 464 |
CCLXXII | 468 |
CCLXXIII | 469 |
CCLXXIV | 475 |
479 | |
481 | |
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Common terms and phrases
algebra alphabet average codeword length BCH code binary code binary symmetric channel Boolean polynomial bound C₁ C₂ code of length codeword coefficients columns compute Corollary cyclic code cyclotomic decision error decision scheme decoding defined Definition Let denote designed distance encoding scheme entropy equations equivalent Example exercise factor finite field Golay code Goppa code Hamming codes Hence idempotent implies inequality input distribution irreducible polynomial Lemma linear code matrix H MDS code minimal polynomial minimum distance modulo multiple narrow sense BCH Noisy Coding Theorem nonzero output P₁ parameters parity check matrix polynomial f(x polynomials of degree positive integer primitive n-th root probability distribution probability of error Proof q-ary quadratic residue quadratic residue codes random variables Reed-Muller code Reed-Solomon code root of f(x root of unity rows Show splitting field string Suppose syndrome Theory vector space zeros