Algebraic Codes on Lines, Planes, and Curves: An Engineering by Richard E. Blahut

By Richard E. Blahut

Algebraic geometry is frequently hired to encode and decode indications transmitted in communique structures. This booklet describes the elemental rules of algebraic coding idea from the viewpoint of an engineer, discussing a few functions in communications and sign processing. The crucial idea is that of utilizing algebraic curves over finite fields to build error-correcting codes. the latest advancements are provided together with the idea of codes on curves, with no using unique arithmetic, substituting the serious idea of algebraic geometry with Fourier remodel the place attainable. the writer describes the codes and corresponding interpreting algorithms in a way that enables the reader to guage those codes opposed to useful functions, or to aid with the layout of encoders and decoders. This ebook is correct to practising communique engineers and people all in favour of the layout of latest communique structures, in addition to graduate scholars and researchers in electric engineering.

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To formulate the algorithm, we will choose one representative b from each conjugacy class and decompose v as the sum of vectors v(b) , v= b (b) where the vector v(b) has nonzero component vi only if i is an element of the conjugacy class of b, which is Ab = {b, pb, p2 b, . . , pr−1 b}, where r is the number of elements in the conjugacy class of b. Thus, vi(b) = vi 0 i ∈ Ab i ∈ Ab . 40 Sequences and the One-Dimensional Fourier Transform Then V (b) , V = b where V (b) is the Fourier transform of v(b) .

One reason for using a finite field (rather than the real field) in an engineering problem is to eliminate problems of round-off error and overflow from computations. However, the arithmetic of a finite field is not well matched to everyday computations. This is why finite fields are most frequently found in those engineering applications in which the computations are introduced artificially as a way of manipulating bits for some purpose such as error control or cryptography. 2 The Fourier transform The (discrete) Fourier transform, when defined in the complex field, is a fundamental tool in the subject of signal processing; its rich set of properties is part of the engineer’s workaday intuition.

1 1 1 .. z ij ... ⎤ ⎡ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ v0 (z) v1 (z) .. ⎤ ⎥ ⎥ ⎥ ⎦ v254 (z) V254 (z) is a 255-point Fourier transform over GF(256). Each component consists of eight bits, represented as a polynomial over GF(2), and powers of z are reduced by using z 8 = z 4 + z 3 + z 2 + 1. (9) Q(16) . The polynomial p(z) = z 16 + 1 is irreducible over Q. Modulo z 16 + 1 multiplication is reduced by setting z 16 = −1. An element of Q(16) may be thought of as a “supercomplex” rational with sixteen parts (instead of two parts).

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