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McEliece R.J. — Finite Fields for Computer Scientists and Engineers
McEliece R.J. — Finite Fields for Computer Scientists and Engineers

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Название: Finite Fields for Computer Scientists and Engineers

Автор: McEliece R.J.


The theory of finite fields is of central importance in engineering and computer science, because of its applications to error-correcting codes, cryptography, spread-spectrum communications, and digital signal processing. Though not inherently difficult, this subject is almost never taught in depth in mathematics courses, (and even when it is the emphasis is rarely on the practical aspect). Indeed, most students get a brief and superficial survey which is crammed into a course on error-correcting codes. It is the object of this text to remedy this situation by presenting a thorough introduction to the subject which is completely sound mathematically, yet emphasizes those aspects of the subject which have proved to be the most important for applications.

Язык: en

Рубрика: Технология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1986

Количество страниц: 207

Добавлена в каталог: 24.11.2013

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Предметный указатель
Antilogarithxns      27
Associate      13 17
Autocorrelation function      156 167
Basis, dual      110
Berlekamp's bit serial multiplication circuits      110ff
Berlekamp's polynomial factorization algorithm      84ff
Binomial coefficients      44 146
Binomial theorem      45 52
Blumer, A.      119
Calculus, freshman      57
Characteristic equation, of linear recursion      124
Chevalley — Warning, Theorem of      182
complex numbers      23 24 70 76
Conjugates      46
Correlation, between two sequences      155
Crosscorrelation function      171—172
Crosstalk      170
Cyclotomic cosets      91
Cyclotomic polynomials      76ff
Decimal, repeating      53
Decimation      162
Degree, of an element in a finite field      47
Derivative, formal      57 72
Distribution problems      137ff
Division algorithm      24
Division, synthetic      24 93
Divisor, proper      14
Domain, integral      3
Euclid      3
Euclid's algorithm, could be taught to junior high school students      6
Euclid's algorithm, extended version of      9
Euclid's algorithm, statement of      7
Euclidean domain, defintion of      3
Euclidean domain, examples of      4
Euler Product technique      58
Euler's $\phi$ function, definition of      33
Euler's $\phi$ function, formulas for      65
Fact, a curious      7 12
Factorization, trivial      13 17
Factorization, unique factorization theorem      15
Fibonacci numbers      7 11 123—125 131 138 141 142 149
Field with four elements      1
Field with one element      2
Field with p elements      1 22
Field, characteristic of      30
Field, definition of      1
Field, finite, existence of      67
Field, infinite are uninteresting      1
Field, uniqueness of      69
Gauss's algorithm for finding primitive roots      38 52
Gaussian integers      4 10 14 17 28
GCD      see "Greatest common divisor"
Generating functions      58
Gold sequences      196 200
Greatest common divisor, computationally clumsy algorithm for finding      16
Greatest common divisor, definition of      4
Greatest common divisor, expressed as a linear combination of things      5
Hilbert's algorithm for solving $x^{q}-x=\alpha$      104ff
Initial conditions, for linear recurrence relation      123
Junior high school algorithm for finding gcd's      16
Kloosterman sum      174
Lagrange's theorem      31
Linear recurrences      123ff
Linear recurrences, characteristic polynomial      127
Linear recurrences, cycles in equivalent solutions to      134
Linear recurrences, cyclic equivalence of solutions to      132
Logarithms      27
m-gram      152
M-sequences      151ff
m-sequences, canonical cyclic shift of      160
m-sequences, crosscorrelation between two, Big Theorem about      193
m-sequences, cycle-and-add property of      159
m-sequences, number of different      161
m-sequences, run-distribution properties of      154
Maximal-length shift register sequences      see "m-sequences"
Moebius function      62ff
Moebius inversion      60ff
Norm, definition of      97
Norm, great, lesser, and relative      100
Odd crosscorrelation function      172
Order, of an element in a finite field      31
Parity tree      113
Period, reduced, of a sequence      137
PN sequences      see "m-sequences"
Polynomials in several variables      182
Polynomials, characteristic, with repeated roots      145 149
Polynomials, interpolation      182
Polynomials, irreducible      14
Polynomials, minimal      41ff
Polynomials, number of irreducible of degree d      57 66
Polynomials, period of      130
Polynomials, primitive      43 151
Polynomials, reciprocal, coefficients of      94
Prime, in Euclidean domain      13
Prime, relatively      14
Primitive element      112n.
Primitive root      37
Primitive root, mod n      83
Projective cyclic equivalence      134.
Pseudo randomness properties, of m-sequences      152ff
Quadratic equations, solution of in characteristic      2 105ff
Quadratic forms      179ff
Quadratic forms, nonsingular      179
Quadratic forms, rank of      185
Quadratic forms, representing zero      179
Signature sequences, used in multi-user communication      169
Subfield      30 70
Trace, definition of      97
Trace, great, lesser and relative      100
Unit, in a Euclidean domain      13 17
Vandermonde matrix      140
Whiting, D.      118
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