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Abstract Algebra: An Introduction
Found in: Page 471
Abstract Algebra: An Introduction

Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

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Short Answer

Question 12: If is a generator polynomial of a BCH code in 2xxn-1, prove that divides xn-1. [Hint: Exercises 11 and 8(b).]


It has been proved that g(x) dividesxn-1 .

See the step by step solution

Step by Step Solution

Step 1: Consider the results of 

Exercise 11: Let K=Z2αbe a finite field of order 2r, whose multiplicative group is generated by . For each i, letmix be the polynomial of over . If ,n=2r-1 then eachmix divides .

Exercise 8: Ifm1x,m2x,...,mkx are distinct monic irreducible inFx such that eachmix divides f(x), then g(x): m1x,m2x,...,mkxdivides f(x)

Step 2: Write the given data

Given that g(x)is a generator polynomial of a BCH code in 2xxn-1.

Step 3: Determine the Proof

Here,gx is a product of distinct minimal polynomialsmix .

According to Exercise 11, mixdivides xn-1

This impliesgx divides xn-1.

Hence, the result.

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