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# chap4 - Chapter 4 Linear Cyclic Codes 4.1 Definition of...

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Chapter 4 Linear Cyclic Codes 4.1 Definition of Cyclic Code An ( n, k ) linear code C is called a cyclic code if any cyclic shift of a codeword is another codeword. That is, if C c , , c , c c 1 - n 1 0 = ) ( L then C c , , c , c , c c 2 - n 1 0 1 - n 1 = ) ( ) ( L Cyclic structure makes the encoding and syndrome computation very easy. In polynomial form 1 - n 1 - n 2 2 1 0 x c x c x c c x c + + + + = L ) ( 1 - n 2 - n 2 1 0 1 - n 1 x c x c x c c x c + + + + = L ) ( ) (

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4.2 Generator Polynomial Every nonzero code polynomial c ( x ) in C must have degree at least n-k but not greater than n-1 . There exists one and only one nonzero generator polynomial g ( x ) for a cyclic code. Uniqueness : Suppose g ( x ) is not unique, then there would exist another such code polynomial of the same degree of the form, k - n 1 - k - n 1 - k - n 2 2 1 x x g' x g' x g' 1 x g' + + + + + = L ) ( . Thus one obtains the code polynomial 1 - k - n 1 - k - n 1 - k - n 1 1 x g g' x g g' x g" ) ( ) ( + + = L ) ( . Polynomial ) ( x g" corresponds to the word ) ) ( 0 , g g' , , g g' , g g' 0, 1 - k - n 1 - k - n 2 2 1 1 ( L . Since k - n g" deg < ) ( , this contradicts the minimality of g ( x ). Therefore g ( x ) must be unique.
Every code polynomial c ( x ) is divisible by g ( x ), i.e. a multiple of g ( x ).

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4.3 Selecting the Parameters of a Cyclic Code It can be shown that the generator polynomial g ( x ) of an ( n, k ) cyclic code is always a polynomial factor of the polynomial 1 - x n , (or 1 x n + ). This means that any factor of 1 - x n of degree n-k is a possible generator of an ( n, k ) cyclic code. Since g ( x ) divides 1 - x n , it follows that ) ( ) ( x g x h 1 - x n = where k k 2 2 1 0 x h x h x h h x h + + + + = L ) ( where 1 h h k 0 = = i.e. k 1 - k 1 - k 2 2 1 x x h x h x h 1 x h + + + + + = L ) ( . h ( x ) is called the parity polynomial of an ( n, k ) cyclic code. Since ) ( ) ( x g x h 1 - x n = , one has the important relation ) ( ) ( ) ( 1 - x mod 0 x g x h n = .
Define message polynomial as 1 - k 1 - k 2 2 1 0 x m x m x m m x m + + + + = L ) ( Clearly, the product m ( x ) g ( x ) is the polynomial that represents the codeword polynomial of degree n-1 or less. An ( n, k ) cyclic code is completely specified by the monic generator polynomial k - n 1 - k - n 1 - k - n 1 x x g x g 1 x g + + + + = L ) ( The codeword polynomial is expressed as 1 - n 1 - n 2 2 1 0 x c x c x c c x c + + + + = L ) ( . 1 - n 2 - n 2 1 0 1 - n 1 x c x c x c c x c + + + + = L ) ( ) ( .

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c ( x ) and ) ( ) ( x c 1 are related by the formula ) ( ) ( ) ( ) ( 1 - x mod x xc x c n 1 = This follows from the fact that since both c ( x ) and ) ( ) ( x c 1 are code polynomials and ) ( ) ( ) ( ) ( ) ( 1 - x c x c 1 - x c x c x c c c - x c x c c x c x c x xc n 1 - n 1 n 1 - n 1 - n 2 - n 0 1 - n 1 - n n 1 - n 0 1 - n n 1 - n 0 + = + + + + = + + + = + + = L L L Now, ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( x g x m 1 - x mod x g x m x x c i n i i = = This identity proves that if any codeword is cyclically shifted i times, that another codeword in the cyclic code C is formed. Thus, any set of k q codewords which possesses the cyclic property is generated by some generator polynomial g ( x ).
4.4 Encoding of Cyclic Codes Consider an ( n, k ) cyclic code with generator polynomial g ( x ). Suppose ) ( 1 - k 1 0 m , , m , m m L = is the message to be encoded. 1 - k 1 - k 1 0 x m x m m x m + + + = L ) ( .

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chap4 - Chapter 4 Linear Cyclic Codes 4.1 Definition of...

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