Error Control Coding (2E, Lin and Costello)- Solutions Manual

Error Control Coding (2E, Lin and Costello)- Solutions Manual

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Chapter 2 2.3 Since m is not a prime, it can be factored as the product of two integers a and b , m = a · b with 1 < a,b < m . It is clear that both a and b are in the set { 1 , 2 , ··· ,m - 1 } . It follows from the definition of modulo- m multiplication that a ¡ b = 0 . Since 0 is not an element in the set { 1 , 2 , - 1 } , the set is not closed under the modulo- m multiplication and hence can not be a group. 2.5 It follows from Problem 2.3 that, if m is not a prime, the set { 1 , 2 , - 1 } can not be a group under the modulo- m multiplication. Consequently, the set { 0 , 1 , 2 , - 1 } can not be a field under the modulo- m addition and multiplication. 2.7 First we note that the set of sums of unit element contains the zero element 0 . For any 1 ‘ < λ , X i =1 1 + λ - X i =1 1 = λ X i =1 1 = 0 . Hence every sum has an inverse with respect to the addition operation of the field GF ( q ) . Since the sums are elements in GF ( q ) , they must satisfy the associative and commutative laws with respect to the addition operation of GF ( q ) . Therefore, the sums form a commutative group under the addition of GF ( q ) . Next we note that the sums contain the unit element 1 of GF ( q ) . For each nonzero sum X i =1 1 with 1 ‘ < λ , we want to show it has a multiplicative inverse with respect to the multipli- cation operation of GF ( q ) . Since λ is prime, and λ are relatively prime and there exist two 1
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integers a and b such that a · + b · λ = 1 , (1) where a and λ are also relatively prime. Dividing a by λ , we obtain a = + r with 0 r < λ. (2) Since a and λ are relatively prime, r 6 = 0 . Hence 1 r < λ Combining (1) and (2), we have · r = - ( b + k‘ ) · λ + 1 Consider X i =1 1 · r X i =1 1 = · r X i =1 1 = - ( b + k‘ ) · λ X i =1 +1 = ( λ X i =1 1)( - ( b + k‘ ) X i =1 1) + 1 = 0 + 1 = 1 . Hence, every nonzero sum has an inverse with respect to the multiplication operation of GF ( q ) . Since the nonzero sums are elements of GF ( q ) , they obey the associative and commutative laws with respect to the multiplication of GF ( q ) . Also the sums satisfy the distributive law. As a result, the sums form a field, a subfield of GF ( q ) . 2.8 Consider the finite field GF ( q ) . Let n be the maximum order of the nonzero elements of GF ( q ) and let α be an element of order n . It follows from Theorem 2.9 that n divides q - 1 , i.e. q - 1 = k · n. Thus n q - 1 . Let β be any other nonzero element in GF ( q ) and let e be the order of β . 2
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Suppose that e does not divide n . Let ( n,e ) be the greatest common factor of n and e . Then e/ ( ) and n are relatively prime. Consider the element β ( n,e ) This element has order e/ ( ) . The element αβ ( n,e ) has order ne/ ( ) which is greater than n . This contradicts the fact that n is the maximum order of nonzero elements in GF ( q ) . Hence e must divide n . Therefore, the order of each nonzero element of GF ( q ) is a factor of n . This implies that each nonzero element of GF ( q ) is a root of the polynomial X n - 1 .
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This note was uploaded on 05/11/2011 for the course ELECTRONIC 005 taught by Professor Haghbin during the Spring '10 term at Sharif University of Technology.

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Error Control Coding (2E, Lin and Costello)- Solutions Manual

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