By Richard E. Blahut
Algebraic geometry is usually hired to encode and decode indications transmitted in conversation platforms. This e-book describes the basic ideas of algebraic coding thought from the point of view of an engineer, discussing a couple of purposes in communications and sign processing. The critical thought is that of utilizing algebraic curves over finite fields to build error-correcting codes. the newest advancements are provided together with the speculation of codes on curves, with no using particular arithmetic, substituting the serious thought of algebraic geometry with Fourier remodel the place attainable. the writer describes the codes and corresponding interpreting algorithms in a fashion that permits the reader to guage those codes opposed to useful functions, or to aid with the layout of encoders and decoders. This e-book is appropriate to practising communique engineers and people interested in the layout of latest communique platforms, in addition to graduate scholars and researchers in electric engineering.
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Extra resources for Algebraic Codes on Lines, Planes, and Curves
Given polynomial V (x), a connection polynomial for the sequence of coefficients of V (x) in GF(q)[x] need not be equal to a locator polynomial for V (x) in GF(q)[x]/ xn − 1 , and this is why we use different names. However, we shall see that, in cases of interest to us, they are the same polynomial. 1 The cyclic complexity and the acyclic complexity of a sequence of blocklength n are equal if the cyclic complexity is not larger than n/2. Proof: This is a simple consequence of the agreement theorem.
N − 1, where 0 = 1. The left side of this equation can be interpreted as the set of coefficients of a polynomial product modulo xn − 1. Translated into the language of polynomials, the equation becomes (x)V (x) = 0 (mod xn − 1), with n−1 V (x) = Vj xj . 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).
There is a Fourier transform of blocklength 32 in the field Q(16) . This is because z 16 = −1 (mod z 16 + 1), so the element z has order 32. This Fourier transform takes a vector of length 32 into another vector of length 32. Components of the vector are polynomials of degree 15 over Q. The Fourier transform has the form 31 Vj (z) = z ij vi (z) i=0 (mod z 16 + 1). 12 Sequences and the One-Dimensional Fourier Transform We can think of this as an operation on a 32 by 16 array of rational numbers to produce another 32 by 16 array of rational numbers.