By F. Oggier, E. Viterbo, Frederique Oggier
Algebraic quantity idea is gaining an expanding impression in code layout for plenty of diverse coding purposes, reminiscent of 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 instrument. the final framework has been constructed within the final ten years and many specific code buildings according to algebraic quantity conception are actually to be had. Algebraic quantity conception and Code layout for Rayleigh Fading Channels offers an outline of algebraic lattice code designs for Rayleigh fading channels, in addition to an academic advent to algebraic quantity thought. the fundamental proof of this mathematical box are illustrated by means of many examples and via machine algebra freeware on the way to make it extra obtainable to a wide viewers. This makes the e-book compatible to be used through scholars and researchers in either arithmetic and communications.
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Additional info for Algebraic Number Theory and Code Design for Rayleigh Fading Channels (Foundations and Trends in Communications and Information Theory)
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.
Let K/Q and L/Q be two ﬁeld extensions of Q. , that ﬁxes Q. Recall that if A and B are rings, a ring homomorphism is a map ψ : A → B that satisﬁes, for all a, b ∈ A (1) ψ(a + b) = ψ(a) + ψ(b) (2) ψ(a · b) = ψ(a) · ψ(b) (3) ψ(1) = 1. 11. A Q–homomorphism ϕ : K → C is called an embedding of K into C. Note that the embedding is an injective map, so that we can really understand it as a way of representing elements of K as complex numbers. 4. [45, p. 41] Let K = Q(θ) be a number ﬁeld of degree n over Q.
If x ∈ OK , we have N (x) and Tr(x) ∈ Z. √ Let us come back to the example of Q( 2), and illustrate these new √ deﬁnitions. √ The roots of the minimal polynomial X 2 − 2 are θ1 = 2 and θ2 = − 2. 2. Integral Basis and Canonical Embedding 47 √ √ and for x ∈ Q( 2), x = a + b 2, a, b ∈ Q √ √ √ √ σ1 (a + b 2) = a + b 2 and σ2 (a + b 2) = a − b 2 . The norm of x is N (x) = σ1 (x)σ2 (x) = a2 − 2b2 , while its trace is Tr(x) = σ1 (x) + σ2 (x) = 2a. These ﬁeld embeddings enable to deﬁne a ﬁrst invariant of a number ﬁeld, that is a property of the ﬁeld that does not depend on the way it is represented.
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