Finite Fields for Computer Scientists and EngineersThis book developed from a course on finite fields I gave at the University of Illinois at Urbana-Champaign in the Spring semester of 1979. The course was taught at the request of an exceptional group of graduate students (includ ing Anselm Blumer, Fred Garber, Evaggelos Geraniotis, Jim Lehnert, Wayne Stark, and Mark Wallace) who had just taken a course on coding theory from me. The theory of finite fields is the mathematical foundation of algebraic coding theory, but in coding theory courses there is never much time to give more than a "Volkswagen" treatment of them. But my 1979 students wanted a "Cadillac" treatment, and this book differs very little from the course I gave in response. Since 1979 I have used a subset of my course notes (correspond ing roughly to Chapters 1-6) as the text for my "Volkswagen" treatment of finite fields whenever I teach coding theory. There is, ironically, no coding theory anywhere in the book! If this book had a longer title it would be "Finite fields, mostly of char acteristic 2, for engineering and computer science applications. " It certainly does not pretend to cover the general theory of finite fields in the profound depth that the recent book of Lidl and Neidereitter (see the Bibliography) does. |
Contents
| 1 | |
| 2 | |
| 3 | |
Building Fields from Euclidean Domains | 23 |
Abstract Properties of Finite Fields | 53 |
Finite Fields Exist and are Unique | 57 |
Factoring Polynomials over Finite Fields | 95 |
Trace Norm and BitSerial Multiplication | 99 |
Linear Recurrences over Finite Fields | 149 |
The Theory of mSequences | 167 |
Crosscorrelation Properties of mSequences | 175 |
| 201 | |
| 203 | |
| 204 | |
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Common terms and phrases
b₁ calculation characteristic polynomial coefficients common divisor complex numbers compute conjugate consider Corollary cosets crosscorrelation function cyclic equivalence cyclotomic polynomials decimation defined denote distinct dual coordinates element of F elements of order equation equivalence classes Euclid's algorithm Euclidean domain exactly Example exists fact Fibonacci numbers field F finite field formula Gaussian integers gcd's gcd(a gcd(e GF 2m GF(q GF(qm given Hence integral domain irreducible factors irreducible polynomials Lemma linear combination linear recurrence m-grams m-sequence of length minimal polynomial mod f(x monic multiplication N₁ nonsingular nonzero elements nonzero solutions number of elements number of solutions ord(a period polynomial f(x polynomial of degree powers primitive polynomial primitive root Problems for Chapter proof of Theorem prove quadratic form relatively prime satisfies sequence st Show solutions to 9.13 subfield Theorem 6.1 Tr(a Tr(a² Tr(x trace unique values variables α²