By Richard E. Blahut
Algebraic geometry is usually hired to encode and decode signs transmitted in communique structures. This ebook describes the basic rules of algebraic coding idea from the point of view of an engineer, discussing a couple of functions in communications and sign processing. The relevant inspiration is that of utilizing algebraic curves over finite fields to build error-correcting codes. the latest advancements are provided together with the speculation of codes on curves, with out using designated arithmetic, substituting the serious concept 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 functional functions, or to aid with the layout of encoders and decoders. This booklet is proper to working towards verbal exchange engineers and people serious 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: An Engineering Approach
The ring of univariate polynomials modulo xn − 1, denoted F[x]/ xn − 1 or F ◦ [x], is an example of a quotient ring. In the quotient ring F[x]/ p(x) , which consists of the set of polynomials of degree smaller than the degree of p(x), the result of a polynomial product is found by ﬁrst computing the polynomial product in F[x], then reducing to a polynomial of degree less than the degree of p(x) by taking the remainder modulo p(x). In F[x]/ xn − 1 , this remainder can be computed by repeated applications of xn = 1.
I=0 (9) Cyclic decimation. Write the spectral index j in terms of a vernier index j and a coarse index j : j =j +nj ; j = 0, . . , n − 1; j = 0, . . , n − 1. Then vn i = 1 n 1 = n n −1 n −1 ω−n i ( j +n j ) ω−n ij Vj +n j j =0 j =0 n −1 n −1 ω−n nij Vj +n j . j =0 j =0 Because ωn = 1, the second term in ω equals 1. Then vn i = 1 n n −1 j =0 γ −i j 1 n n −1 Vj +n j j =0 where γ = ωn has order n . , 16 Sequences and the One-Dimensional Fourier Transform (10) Poisson summation. The left side is the direct computation of the zero component of the Fourier transform of the decimated sequence.
The Fourier transform can also be understood as the evaluation of a polynomial. The polynomial representation of the vector v = [vi | i = 0, . . , n − 1] is the polynomial n−1 v (x) = vi xi . i=0 The evaluation of the polynomial v (x) at β is the ﬁeld element v (β), where n−1 v (β) = vi β i . i=0 The Fourier transform, then, is the evaluation of the polynomial v (x) on the n powers of ω, an element of order n. Thus component Vj equals v (ω j ) for j = 0, . . , n − 1. If F is the ﬁnite ﬁeld GF(q) and ω is a primitive element, then the Fourier transform evaluates v (x) at all q − 1 nonzero elements of the ﬁeld.