By F. Oggier, E. Viterbo, Frederique Oggier

ISBN-10: 1933019077

ISBN-13: 9781933019079

ISBN-10: 1933019557

ISBN-13: 9781933019550

Algebraic quantity thought is gaining an expanding influence in code layout for plenty of various coding purposes, resembling unmarried antenna fading channels and extra lately, MIMO structures. prolonged paintings has been performed on unmarried antenna fading channels, and algebraic lattice codes were confirmed to be an efficient software. the final framework has been constructed within the final ten years and many specific code structures in line with algebraic quantity concept at the moment are on hand. Algebraic quantity idea and Code layout for Rayleigh Fading Channels offers an summary of algebraic lattice code designs for Rayleigh fading channels, in addition to an instructional creation to algebraic quantity idea. the elemental proof of this mathematical box are illustrated via many examples and by means of laptop algebra freeware as a way to make it extra obtainable to a wide viewers. This makes the e-book appropriate to be used by way of scholars and researchers in either arithmetic and communications.

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Extra resources for Algebraic Number Theory And Code Design For Rayleigh Fading Channels (Foundations and Trends in Communications and Information The)

Example text

An ideal I of R is principal if it is of the form I = (x) = (x)R = {xy, y ∈ R}, x ∈ I. 4. If R = Z, we have that nZ is a principal ideal of Z for all n. We can define the norm of an ideal. In the case where the ideal is principal, it is directly related to the norm of a generator of the ideal. 4. 18. Let I = (x)OK be a principal ideal of OK . Its norm is defined by N (I) = |N (x)|. It can be shown that all ideals of OK have a Z-basis of n elements. 10. [45, p. 121] Every ideal I = {0} of OK has a Z-basis {ν1 , .

Let {ω1 , ω2 , . . ωn } be an integral basis of K. The discriminant of K is defined as dK = det[(σj (ωi ))ni,j=1 ]2 . It can be shown that the discriminant is independent of the choice of a basis [43]. 6. [45, p. 51] The discriminant dK of a number field belongs to Z. √ Let us compute the discriminant dK of the field Q( 5). Applying √ the two Q-homomorphisms to the integral basis {ω1 , ω2 } = {1, (1+ 5)/2}, we obtain dK = det σ1 (1) σ2 (1) √ √ 1+ 5 σ1 ( 2 ) σ2 ( 1+2 5 ) 2 = det 1√ 1+ 5 2 1√ 1− 5 2 2 =5.

In [51] the explicit geometric interpretation in terms of Sphere Decoder was shown. In the following, it will be useful to think of the lattice Λ as the result of a linear transformation, defined by the matrix M : Rn → Rn , when applied to the Zn –lattice. So Λ can be seen as a skewed version of the Zn –lattice. The problem to solve is the following: min r − x x∈Λ 2 = min w∈r−Λ w 2 . 1) that is, we search for the shortest vector w in the translated lattice r − Λ in the n-dimensional Euclidean space Rn .

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Algebraic Number Theory And Code Design For Rayleigh Fading Channels (Foundations and Trends in Communications and Information The) by F. Oggier, E. Viterbo, Frederique Oggier

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