By Richard E. Blahut
Algebraic geometry is frequently hired to encode and decode signs transmitted in verbal exchange platforms. This e-book describes the basic rules of algebraic coding idea from the point of view of an engineer, discussing a couple of purposes in communications and sign processing. The crucial thought is that of utilizing algebraic curves over finite fields to build error-correcting codes. the newest advancements are awarded together with the speculation of codes on curves, with no using specified arithmetic, substituting the serious thought of algebraic geometry with Fourier rework the place attainable. the writer describes the codes and corresponding interpreting algorithms in a fashion that permits the reader to judge those codes opposed to useful purposes, or to aid with the layout of encoders and decoders. This booklet is correct to practising verbal exchange engineers and people fascinated by the layout of recent communique platforms, 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
Hence for each power of 2 up to 216 , GF(216 + 1) contains a Fourier transform of blocklength n equal to that power of 2. (6) GF((217 − 1)2 ). This ﬁeld is constructed as an extension of GF(217 − 1), using a polynomial of degree 2 that is irreducible over GF(217 − 1). An element ω of order n exists in the extension ﬁeld if n divides (217 − 1)2 − 1 = 218 (216 − 1). In particular, for each power of 2 up to 218 , GF((217 − 1)2 ) contains a Fourier transform of blocklength equal to that power of 2.
3 (Hartmann–Tzeng bound) Suppose that b and n are coprime. The only vector v of blocklength n of weight d − 1 or less, whose spectral components satisfy V((a+ 1 +b 2 )) 1 =0 2 = 0, . . , d − 2 − s = 0, . . , s, is the all-zero vector. Proof: This bound is a special case of the Roos bound, which is given next. Notice that the Hartmann–Tzeng bound is based on s+1 uniformly spaced substrings of zeros in the spectrum, each substring of length d −1−s. The Roos bound, given next, allows the evenly spaced repetition of these s + 1 substrings of zeros to be interrupted by some nonzero substrings, as long as there are not too many such nonzero substrings.
Qr − 1, then it also holds for j = qr. q Proof: We shall give two expressions for the same term. By assumption, Vj = V((qj)) . The ﬁrst expression is given by q L q Vr = − i Vr−i L L q q i Vr−i =− i=1 q i Vq(r−i) . =− i=1 i=1 To derive the second expression, embed the linear recursion into itself to obtain ⎡ ⎤ L Vqr = − L k Vqr−k = − k1 ⎣− k1 =1 k=1 L k2 Vqr−k1 −k2 ⎦ k2 =1 L L L ··· = (−1)q k1 =1 k2 =1 k1 k2 ··· kq Vqr−k1 −k2 −···−kq . kq =1 The ﬁnal step of the proof is to collapse the sum on the right, because, unless k1 = k2 = k3 = · · · = kq , each term will recur in multiples of the ﬁeld characteristic p, and each group of p identical terms adds to zero modulo p.