By Richard E. Blahut
Algebraic geometry is frequently hired to encode and decode indications transmitted in communique platforms. This e-book describes the basic ideas of algebraic coding idea from the point of view of an engineer, discussing a couple of functions in communications and sign processing. The central thought is that of utilizing algebraic curves over finite fields to build error-correcting codes. the latest advancements are offered together with the speculation of codes on curves, with out using exact arithmetic, substituting the serious idea of algebraic geometry with Fourier remodel the place attainable. the writer describes the codes and corresponding deciphering algorithms in a fashion that enables the reader to judge those codes opposed to useful functions, or to aid with the layout of encoders and decoders. This publication is correct to practising verbal exchange engineers and people eager about the layout of recent verbal exchange platforms, in addition to graduate scholars and researchers in electric engineering.
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Extra info for Algebraic Codes on Lines, Planes, and Curves
J=0 In the inverse Fourier transform domain, the cyclic convolution becomes λi vi = 0, where λi and vi are the ith components of the inverse Fourier transform. Thus λi must be zero whenever vi is nonzero. In this way, the connection polynomial (x) that achieves the cyclic complexity locates, by its zeros, the nonzeros of the polynomial V (x). To summarize, the connection polynomial is defined by its role in the linear recursion. If the sequence it produces is periodic, however, then it has another property.
Proof: This is a simple consequence of the agreement theorem. The acyclic complexity is clearly not larger than the cyclic complexity. Thus, by assumption, the recursions for the two cases are each of length at most n/2, and they agree at least until the nth symbol of the sequence. Hence, by the agreement theorem, they continue to agree thereafter. The linear complexity property can be combined with the cyclic permutation property of the Fourier transform to relate the recursions that produce two periodic sequences that are related by a cyclic permutation.
2 If for any sequence V0 , V1 , . . , Vn−1 over a field of characteristic 2, satisfying Vj2 = V((2j)) , and for any linear recursion ( (x), L), L Vj = − j = L, . . , 2r − 1, i Vj−i i=1 then L V2r = − i V2r−i . i=1 Proof: By assumption, V2r = Vr2 . The proof consists of giving two expressions for the same term. First, using 1 + 1 = 0 in a field of characteristic 2, we have that 2 L Vr2 = i Vr−i i=1 L = L 2 2 i Vr−i i=1 = 2 i V2r−2i . i=1 Second, we have that L V2r = − L k V2r−k k=1 = L k i V2r−k−i .