By F. Oggier, E. Viterbo, Frederique Oggier

ISBN-10: 1933019077

ISBN-13: 9781933019079

Algebraic quantity conception is gaining an expanding effect in code layout for lots of diverse coding functions, akin to unmarried antenna fading channels and extra lately, MIMO platforms. prolonged paintings has been performed on unmarried antenna fading channels, and algebraic lattice codes were confirmed to be an efficient software. the overall framework has been constructed within the final ten years and many particular code structures according to algebraic quantity idea at the moment are on hand. Algebraic quantity idea and Code layout for Rayleigh Fading Channels offers an outline of algebraic lattice code designs for Rayleigh fading channels, in addition to an educational creation to algebraic quantity concept. the fundamental evidence of this mathematical box are illustrated by means of many examples and via desktop algebra freeware that allows you to make it extra obtainable to a wide viewers. This makes the publication 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 Theory)**

**Sample text**

1). Furthermore, we will see that the minimum product distance can be easily computed in this case. Let K be a totally real number ﬁeld of degree n, and let Λ(OK ) be an algebraic lattice built over OK . Then its lattice generator simpliﬁes to ⎞ ⎛ σ1 (ω1 ) σ2 (ω1 ) . . σn (ω1 ) ⎜ σ1 (ω2 ) σ2 (ω2 ) . . σn (ω2 ) ⎟ ⎟ ⎜ M =⎜ ⎟ . .. ⎠ ⎝ . σ1 (ωn ) σ2 (ωn ) . . σn (ωn ) The product distance of x from 0 is related to the algebraic norm [18]: n n |xj | = dp(n) (0, x) = j=1 |σj (x)| = |N (x)| j=1 TEAM LinG 54 First Concepts in Algebraic Number Theory with x ∈ OK .

The canonical embedding gives a geometrical representation of a number ﬁeld, the one that will serve our purpose. 3 Algebraic Lattices We are now ready to introduce algebraic lattices. 15) establishes a one-to-one correspondence between the elements of an algebraic number ﬁeld of degree n and the vectors of the n-dimensional Euclidean space. The ﬁnal step for constructing an algebraic lattice is given by the following result. 7. [45, p. 155] Let {ω1 , ω2 , . . 3. Algebraic Lattices K. The n vectors vi = σ(ωi ) ∈ Rn , i = 1, .

Xn ) ∈ Z n+1 , xi = 0}. 4 2 −1 0 . . −1 2 −1 0 −1 2 .. . 0 0 0 ... 0 0 0 .. , by maximizing the number of spheres which can ﬁt inside the box. Such arrangements of spheres are called sphere packings. The spheres will not ﬁll all the space in the box and whatever arrangement is chosen at least about 25% of the space remains empty. We call packing density ∆ the percentage of space occupied by the spheres. The above problem can be generalized to higher or lower dimensions, but the optimal or densest sphere packing is only known in dimensions 1 and 2 (Fig.

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

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