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Unformatted text preview: Error Correcting Codes: Combinatorics, Algorithms and Applications (Spring 2011) Lecture N: Intro to Polynomial Fields 2/23/2011 Lecturer: Atri Rudra Scribe: Dan Padgett 1 Polynomials Definition 1.1. Let F q be a finite field of order q . Then a function P ( x ) = ∑ ∞ i =0 p i x i ,p i ∈ F q is called a (univariate) polynomial. For our purposes, we will only consider the finite case; that is, P ( x ) = ∑ d i =0 p i x i ,p i ∈ F q for some integer d > and p d 6 = 0 . Definition 1.2. In this case, we call d the degree of P ( x ) . We notate this by deg( P ) . Let F q [ x ] be the set of polynomials over F q , that is, with coefficients from F q . Let P ( x ) ,Q ( x ) ∈ F q [ x ] be polynomials. Then F q [ x ] is a ring with the following operations: Addition: P ( x ) + Q ( x ) = max(deg( P ) , deg( Q )) X i =0 ( p i + q i ) x i Multiplication: P ( x ) · Q ( x ) i.e. x (1 + x ) = x + x 2 ; (1 + x ) 2 = 1 + 2 x + x 2 = 1 + x 2 with q = 2 ....
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- Spring '11
- Algorithms, Prime number, Complex number, finite field, fQ, Polynomial code