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

By Richard E. Blahut

Algebraic geometry is frequently hired to encode and decode indications transmitted in verbal exchange structures. This booklet describes the elemental rules of algebraic coding conception from the viewpoint of an engineer, discussing a couple of purposes in communications and sign processing. The important inspiration is that of utilizing algebraic curves over finite fields to build error-correcting codes. the latest advancements are awarded together with the speculation of codes on curves, with out using specified arithmetic, substituting the serious conception of algebraic geometry with Fourier rework 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 purposes, or to assist with the layout of encoders and decoders. This booklet is appropriate to practising verbal exchange engineers and people fascinated by the layout of latest communique platforms, in addition to graduate scholars and researchers in electric engineering.

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If the extension field E contains an element ω of order n, then there is a Fourier transform of blocklength n in E, which has the same form as before: n−1 Vj = ωij vi j = 0, . . , n − 1. i=0 Now, however, the vector V has components in the extension field E even if v has components only in the field F. We wish to describe the nature of any vector V in the vector space E n that is the Fourier transform of a vector v in the vector space F n . 1 The vector V over the complex field C is the Fourier transform of a vector v over the real field R if and only if, for all j, Vj∗ = Vn−j .

It should also be noted that if b(x) = a[r] (x), then, in general, b[k] (x) = a[r+k] (x). Hence this useful and well known property of the formal derivative does not carry over to the Hasse derivative. The following theorem gives a property that does follow over. 1 (Hasse) If h(x) is an irreducible polynomial of degree at least 1, then [h(x)]m divides f (x) if and only if h(x) divides f [ℓ] (x) for ℓ = 0, . . , m − 1. 14. 5 Linear complexity of sequences A linear recursion (or recursion) over the field F is an expression of the form L Vj = − j = L, L + 1, .

Hence only trivial Fourier transforms exist in Q or R. To obtain a Fourier transform over R of blocklength larger than 2, one must regard R as embedded into C. There is, however, a multidimensional Fourier transform over Q or R with 2m elements. It uses ω = −1 and a Fourier transform of length 2 on each dimension of a two by two by . . by two m-dimensional array, and it is a nontrivial example of a multidimensional Fourier transform in the fields Q and R. ) 10 Sequences and the One-Dimensional Fourier Transform √ (2) C: ω = e−i2π/n has order n, where i = −1.

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