By Richard E. Blahut

ISBN-10: 0511388608

ISBN-13: 9780511388606

ISBN-10: 0521771943

ISBN-13: 9780521771948

Algebraic geometry is frequently hired to encode and decode indications transmitted in communique platforms. This e-book describes the basic ideas of algebraic coding conception from the point of view of an engineer, discussing a few functions in communications and sign processing. The central suggestion is that of utilizing algebraic curves over finite fields to build error-correcting codes. the latest advancements are awarded together with the idea of codes on curves, with no using precise 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 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 appropriate to working towards communique engineers and people desirous about the layout of latest communique 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**

**Example text**

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.

The set 2 {β, β q , β q , . . , β q r−1 } is called the set of q-ary conjugates of β (or the Galois orbit of β). If γ is a conjugate of β, then β is a conjugate of γ . In general, an element has more than one q-ary conjugate. If an element of GF(qm ) has r q-ary conjugates (including itself), it is an element of the subﬁeld GF(qr ) ⊂ GF(qm ), so r divides m. Thus, under conjugacy, the ﬁeld decomposes into disjoint subsets called conjugacy classes. The term might also be used to refer to the set of exponents on a primitive element of the members of a set of q-ary conjugates.

For no other n is there an ω in Q or R of order n. 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.

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

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