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# GRS - 62 Chapter 5 Generalized Reed-Solomon Codes In 1960...

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Unformatted text preview: 62 Chapter 5 Generalized Reed-Solomon Codes In 1960, I.S. Reed and G. Solomon introduced a family of error-correcting 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 Reed-Solomon 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 α 1 ,...,α n ∈ F . Set v = ( v 1 ,...,v n ) and α = ( α 1 ,...,α n ). For ≤ k ≤ n we define the generalized Reed-Solomon codes generalized Reed-Solomon codes GRS n,k ( α , v ) = { ( v 1 f ( α 1 ) ,v 2 f ( α 2 ) ,...,v n f ( α 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 , α , and v , the various GRS codes enjoy the nice embedding property GRS n,k- 1 ( α , v ) ≤ GRS n,k ( α , v ). If f ( x ) is a polynomial, then we shall usually write f for its associated code- word. This codeword also depends upon α and v ; so at times we prefer to write unambiguously ev α , v ( f ( x )) = ( v 1 f ( α 1 ) ,v 2 f ( α 2 ) ,...,v n f ( α n )) , indicating that the codeword f = ev α , v ( f ( x )) arises from evaluating the poly- nomial f ( x ) at α and scaling by v . (5.1.1) Theorem. GRS n,k ( α , 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 REED-SOLOMON CODES Proof. As by definition the entries in α 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 α , v ( af ( x ) + g ( x )) = a ev α , v ( f ( x )) + ev α , v ( g ( x )) = a f + g . Therefore GRS n,k ( α , 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 0’s in h . As all the v i are nonzero, this equals n minus the number of roots that h ( x ) has among { α 1 ,...,α 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- α 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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GRS - 62 Chapter 5 Generalized Reed-Solomon Codes In 1960...

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