By Richard E. Blahut
Algebraic geometry is usually hired to encode and decode indications transmitted in verbal exchange structures. This publication describes the elemental ideas of algebraic coding conception from the point of view of an engineer, discussing a few purposes in communications and sign processing. The relevant proposal is that of utilizing algebraic curves over finite fields to build error-correcting codes. the newest advancements are offered together with the speculation of codes on curves, with no using certain 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 sensible functions, or to assist with the layout of encoders and decoders. This ebook is suitable to practising verbal exchange engineers and people inquisitive about the layout of latest communique structures, in addition to graduate scholars and researchers in electric engineering.
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Additional info for Algebraic Codes on Lines, Planes, and Curves: An Engineering Approach
The set of all locator polynomials for a given V (x) forms an ideal, called the locator ideal. Because GF(q)[x]/ xn − 1 is a principal ideal ring, meaning that any ideal is generated by a single polynomial of minimum degree, the locator ideal is a principal ideal. All generator polynomials for this ideal have minimum degree and are scalar multiples of any one of them. All elements of the ideal are polynomial multiples of any generator polynomial. It is conventional within the subject of this book to speak of the unique locator polynomial by imposing the requirements that it have minimal degree and the constant term 0 is equal to unity.
Two polynomials v (x) and v (x) over the same ﬁeld can be added by the rule (vi + vi )xi , v (x) + v (x) = i and can be multiplied by the rule vj vi−j xi . v (x)v (x) = i j The division algorithm for univariate polynomials is the statement that, for any two nonzero univariate polynomials f (x) and g(x), there exist uniquely two polynomials Q(x), called the quotient polynomial, and r(x), called the remainder polynomial, such that f (x) = Q(x)g(x) + r(x), and deg r(x) < deg g(x). The reciprocal polynomial of v (x), a polynomial of degree r, is the polynomial v˜ (x) = 0i=r vr−i xi .
D − 2 is zero in at least s + 1 components, and so is nonzero in at most d − s − 2 components. The same is true for the sequence Va+ b+ 2 for each of d − s − 1 values of 2 , so all the missing zeros can be created. 4 provided the new vector is not identically zero. But it is easy to see that the new vector cannot be identically zero unless the original vector has a string of d − 1 consecutive zeros in its spectrum, in which case the BCH bound applies. The ﬁnal bound of this section subsumes all the other bounds, but the proof is less transparent and the bound is not as easy to use.
Algebraic Codes on Lines, Planes, and Curves: An Engineering Approach by Richard E. Blahut