Material de lectura 7 - Information Theory and Coding(66.24...

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Information Theory and Coding (66.24) Cyclic Codes A linear code is said to be cyclic if the i th cyclic rotation is also a codeword of the same code. Recalling groups, a group is said to be cyclic if it can be generated by succesive powers of a given element. A cyclic code can be represented as polynomials defined over a Galois field GF(2 n ). A codeword c = ( c 0 , c 1 , · · · , c n - 1 ) is in the form: c ( X ) = c 0 + c 1 + · · · + c n - 1 X n - 1 c i GF (2) Polynomials over GF(2) can be added (substracted), multiplied, and di- vided in the usual way using binary field arithmetic. A given polynomial f ( X ) over GF(2) having an even number of terms, is divisible by X + 1. Why? Definition 1 A polynomial f ( X ) over GF(2) of degree m is said to be irreducible if it is not divisible by any polynomial over GF (2) of degree less than m but greater than zero. Example: X 3 + X + 1 does not have either 0 or 1 as a root and so is not divisible by X or X +1. Since it is not divisible by any polynomial of degree 1, then it is not divisible by a polynomial of degree 2. Why? Theorem 1 Any irreducible polynomial over GF(2) of degree m di- vides X 2 m - 1 + 1. Definition 2 An irreducible polynomial f ( X ) of degree m is said to be primitive if the smallest positive integer n for which f ( X ) divides X n +1 is n = 2 m - 1. Recall the definition of primitive elements in GF(q). Construction of Galois Fields GF( 2 n ) Given a primitive polynomial of degree m over GF(2) p ( X ), let α be a new element such that p ( α ) = 0. Since p ( X ) divides X 2 n - 1 + 1, then, X 2 n - 1 + 1 = q ( X ) p ( X ) 1
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Replacing: α 2 n - 1 + 1 = q ( α ) p (
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