WE - 124 Chapter 9 Weight and Distance Enumeration The...

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Unformatted text preview: 124 Chapter 9 Weight and Distance Enumeration The weight and distance enumerators record the weight and distance informa- tion for the code. In turn they can be analyzed to reveal properties of the code. The most important result is MacWilliams’ Theorem, which we prove several times. We also prove the related Delsarte Bound and Lloyd’s Theorem. 9.1 Basics The basic definitions are: Definition. Let code C ⊆ F n ( F a field) contain c i codewords of weight i , for i = 1 ,...n . Then the weight enumerator is weight enumerator W C ( z ) = X c ∈ C z w H ( c ) = n X i =0 c i z i ∈ Z [ z ] . The homogeneous weight enumerator is homogeneous weight enumerator W C ( x,y ) = x n W C ( y/x ) = n X i =0 c i x n- i y i ∈ Z [ x,y ] . Actually these definitions make sense whenever the alphabet admits addition, an example of interest being F = Z s . Definition. The distance enumerator of the code A is given by distance enumerator W A ( z ) = | A |- 1 X c , d ∈ A z d H ( c , d ) ∈ Q [ z ] . This can be defined for any alphabet. The notation does not greatly conflict with that above, since the distance enumerator of A equals the weight enumerator 125 126 CHAPTER 9. WEIGHT AND DISTANCE ENUMERATION of A when A is linear. (Indeed, for a code defined over an alphabet admitting addition, we can translate each codeword to the-word and get an associated weight enumerator. The distance enumerator is then the average of these weight enumerators.) One could also define a homogeneous distance enumerator. The basic results are that of MacWilliams: (9.1.1) Theorem. (MacWilliams’ Theorem.) Let C be a [ n,k ] linear code over F s . Set W C ( z ) = n X i =0 c i z i and W C ⊥ ( z ) = n X i =0 c ⊥ i z i . Then (1) W C ( z ) = | C ⊥ |- 1 ∑ n i =0 c ⊥ i (1 + ( s- 1) z ) n- i (1- z ) i , and (2) W C ( x,y ) = | C ⊥ |- 1 W C ⊥ ( x + ( s- 1) y,x- y ) . and its nonlinear relative due to Delsarte: (9.1.2) Theorem. (Delsarte’s Theorem.) Let A be a code in F n with distance enumerator W A ( z ) = ∑ n i =0 a i z i . Define the rational numbers b m by | A |- 1 n X i =0 a i (1 + ( s- 1) z ) n- i (1- z ) i = n X m =0 b m z m . Then b m ≥ , for all m . These two results are related to Lloyd’s Theorem 9.4.9, which states that cer- tain polynomials associated with perfect codes must have integral roots. Lloyd’s Theorem is the most powerful tool available for proving nonexistence results for perfect codes. 9.2 MacWilliams’ Theorem and performance In this section we relate weight enumerators to code performance. This leads to a first proof of MacWilliams’ Theorem. For easy of exposition, we shall restrict ourselves to the consideration of binary linear codes on the BSC( p ) throughout this section. Let C be a binary [ n,k ] linear code. (See Theorem 9.4.8 below for the general case of MacWilliams’ Theorem 9.1.1.) We begin with performance analysis for the binary linear code C on the BSC( p ) under the basic error detection algorithm SS = D : Algorithm D...
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WE - 124 Chapter 9 Weight and Distance Enumeration The...

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