This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full
Document
 Spring '11
 RUDRA
 Algorithms

Click to edit the document details