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Unformatted text preview: 62 Chapter 5 Generalized ReedSolomon Codes In 1960, I.S. Reed and G. Solomon introduced a family of errorcorrecting codes that are doubly blessed. The codes and their generalizations are useful in prac tice, and the mathematics that lies behind them is interesting. In the first sec tion we give the basic properties and structure of the generalized ReedSolomon codes, and in the second section we describe in detail one method of algebraic decoding that is quite efficient. 5.1 Basics Let F be a field and choose nonzero elements v 1 ,...,v n F and distinct elements 1 ,..., n F . Set v = ( v 1 ,...,v n ) and = ( 1 ,..., n ). For k n we define the generalized ReedSolomon codes generalized ReedSolomon codes GRS n,k ( , v ) = { ( v 1 f ( 1 ) ,v 2 f ( 2 ) ,...,v n f ( n ))  f ( x ) F [ x ] k } . Here we write F [ x ] k for the set of polynomial in F [ x ] of degree less than k , a vector space of dimension k over F . For fixed n , , and v , the various GRS codes enjoy the nice embedding property GRS n,k 1 ( , v ) GRS n,k ( , v ). If f ( x ) is a polynomial, then we shall usually write f for its associated code word. This codeword also depends upon and v ; so at times we prefer to write unambiguously ev , v ( f ( x )) = ( v 1 f ( 1 ) ,v 2 f ( 2 ) ,...,v n f ( n )) , indicating that the codeword f = ev , v ( f ( x )) arises from evaluating the poly nomial f ( x ) at and scaling by v . (5.1.1) Theorem. GRS n,k ( , v ) is an [ n,k ] linear code over F with length n  F  . We have d min = n k + 1 provided k 6 = 0 . In particular, GRS codes are MDS codes. 63 64 CHAPTER 5. GENERALIZED REEDSOLOMON CODES Proof. As by definition the entries in are distinct, we must have n  F  . If a F and f ( x ) ,g ( x ) F [ x ] k , then af ( x ) + g ( x ) is also in F [ x ] k ; and ev , v ( af ( x ) + g ( x )) = a ev , v ( f ( x )) + ev , v ( g ( x )) = a f + g . Therefore GRS n,k ( , v ) is linear of length n over F . Let f ( x ) ,g ( x ) F [ x ] k be distinct polynomials. Set h ( x ) = f ( x ) g ( x ) 6 = 0, also of degree less than k . Then h = f g and w H ( h ) = d H ( f , g ). But the weight of h is n minus the number of 0s in h . As all the v i are nonzero, this equals n minus the number of roots that h ( x ) has among { 1 ,..., n } . As h ( x ) has at most k 1 roots by Proposition A.2.10, the weight of h is at least n ( k 1) = n k + 1. Therefore d min n k + 1, and we get equality from the Singleton bound 3.1.14. (Alternatively, h ( x ) = Q k 1 i =1 ( x i ) produces a codeword h of weight n k + 1.) The argument of the previous paragraph also shows that distinct polynomials f ( x ) ,g ( x ) of F [ x ] k give distinct codewords. Therefore the code contains  F  k codewords and has dimension k . 2 The vector v plays little role here, and its uses will be more apparent later....
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This note was uploaded on 06/13/2011 for the course CRYPTO 101 taught by Professor Na during the Spring '11 term at Harding.
 Spring '11
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