The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
Roope Vehkalahti - One of the best experts on this subject based on the ideXlab platform.
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the dmt of real and quaternionic lattice codes and dmt classification of Division Algebra codes
arXiv: Information Theory, 2021Co-Authors: Roope Vehkalahti, Laura LuzziAbstract:In this paper we consider the diversity-multiplexing gain tradeoff (DMT) of so-called minimum delay asymmetric space-time codes. Such codes are less than full dimensional lattices in their natural ambient space. Apart from the multiple input single output (MISO) channel there exist very few methods to analyze the DMT of such codes. Further, apart from the MISO case, no DMT optimal asymmetric codes are known. We first discuss previous criteria used to analyze the DMT of space-time codes and comment on why these methods fail when applied to asymmetric codes. We then consider two special classes of asymmetric codes where the code-words are restricted to either real or quaternion matrices. We prove two separate diversity-multiplexing gain trade-off (DMT) upper bounds for such codes and provide a criterion for a lattice code to achieve these upper bounds. We also show that lattice codes based on Q-central Division Algebras satisfy this optimality criterion. As a corollary this result provides a DMT classification for all Q-central Division Algebra codes that are based on standard embeddings. While the Q-central Division Algebra based codes achieve the largest possible DMT of a code restricted to either real or quaternion space, they still fall short of the optimal DMT apart from the MISO case.
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towards a complete dmt classification of Division Algebra codes
arXiv: Information Theory, 2015Co-Authors: Laura Luzzi, Roope Vehkalahti, Alexander GorodnikAbstract:This work aims at providing new bounds for the diversity multiplexing gain trade-off of a general class of Division Algebra based lattice codes. In the low multiplexing gain regime, some bounds were previously obtained from the high signal-to-noise ratio estimate of the union bound for the pairwise error probabilities. Here these results are extended to cover a larger range of multiplexing gains. The improvement is achieved by using ergodic theory in Lie groups to estimate the behavior of the sum arising from the union bound. In particular, the new bounds for lattice codes derived from Q-central Division Algebras suggest that these codes can be divided into two subclasses based on their Hasse invariants at the infinite places. Algebras with ramification at the infinite place seem to provide better diversity-multiplexing gain tradeoff.
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Division Algebra codes achieve MIMO block fading channel capacity within a constant gap
2015Co-Authors: Laura Luzzi, Roope VehkalahtiAbstract:This work addresses the question of achieving capacity with lattice codes in multi-antenna block fading channels when the number of fading blocks tends to infinity. In contrast to the standard approach in the literature which employs random lattice ensembles, the existence results in this paper are derived from number theory. It is shown that a multiblock construction based on Division Algebras achieves rates within a constant gap from block fading capacity both under maximum likelihood decoding and naive lattice decoding. First the gap to capacity is shown to depend on the discriminant of the chosen Division Algebra; then class field theory is applied to build families of Algebras with small discriminants. The key element in the construction is the choice of a sequence of Division Algebras whose centers are number fields with small root discriminants.
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A new design criterion for spherically-shaped Division Algebra-based space-time codes
2013Co-Authors: Laura Luzzi, Roope VehkalahtiAbstract:This work considers normalized inverse determinant sums as a tool for analyzing the performance of Division Algebra based space-time codes for multiple antenna wireless systems. A general union bound based code design criterion is obtained as a main result. In our previous work, the behavior of inverse determinant sums was analyzed using point counting techniques for Lie groups; it was shown that the asymptotic growth exponents of these sums correctly describe the diversity-multiplexing gain trade-off of the space-time code for some multiplexing gain ranges. This paper focuses on the constant terms of the inverse determinant sums, which capture the coding gain behavior. Pursuing the Lie group approach, a tighter asymptotic bound is derived, allowing to compute the constant terms for several classes of space-time codes appearing in the literature. The resulting design criterion suggests that the performance of Division Algebra based codes depends on several fundamental Algebraic invariants of the underlying Algebra.
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connecting dmt of Division Algebra space time codes and point counting in lie groups
International Symposium on Information Theory, 2012Co-Authors: Roope Vehkalahti, Laura LuzziAbstract:Earlier it was proven by Vehkalahti and Lu how the unit group and diversity-multiplexing gain trade-off (DMT) of Division Algebra-based space-time codes are linked to each other through inverse determinant sums. This work explores this relation further, showing that indeed the density of unit group completely determines the growth of the inverse determinant sum. In particular, in the case of Q(i)-central Division Algebras, the lower bound obtained from the DMT and the upper bound derived from the growth rate of units coincide.
Laura Luzzi - One of the best experts on this subject based on the ideXlab platform.
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the dmt of real and quaternionic lattice codes and dmt classification of Division Algebra codes
arXiv: Information Theory, 2021Co-Authors: Roope Vehkalahti, Laura LuzziAbstract:In this paper we consider the diversity-multiplexing gain tradeoff (DMT) of so-called minimum delay asymmetric space-time codes. Such codes are less than full dimensional lattices in their natural ambient space. Apart from the multiple input single output (MISO) channel there exist very few methods to analyze the DMT of such codes. Further, apart from the MISO case, no DMT optimal asymmetric codes are known. We first discuss previous criteria used to analyze the DMT of space-time codes and comment on why these methods fail when applied to asymmetric codes. We then consider two special classes of asymmetric codes where the code-words are restricted to either real or quaternion matrices. We prove two separate diversity-multiplexing gain trade-off (DMT) upper bounds for such codes and provide a criterion for a lattice code to achieve these upper bounds. We also show that lattice codes based on Q-central Division Algebras satisfy this optimality criterion. As a corollary this result provides a DMT classification for all Q-central Division Algebra codes that are based on standard embeddings. While the Q-central Division Algebra based codes achieve the largest possible DMT of a code restricted to either real or quaternion space, they still fall short of the optimal DMT apart from the MISO case.
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towards a complete dmt classification of Division Algebra codes
arXiv: Information Theory, 2015Co-Authors: Laura Luzzi, Roope Vehkalahti, Alexander GorodnikAbstract:This work aims at providing new bounds for the diversity multiplexing gain trade-off of a general class of Division Algebra based lattice codes. In the low multiplexing gain regime, some bounds were previously obtained from the high signal-to-noise ratio estimate of the union bound for the pairwise error probabilities. Here these results are extended to cover a larger range of multiplexing gains. The improvement is achieved by using ergodic theory in Lie groups to estimate the behavior of the sum arising from the union bound. In particular, the new bounds for lattice codes derived from Q-central Division Algebras suggest that these codes can be divided into two subclasses based on their Hasse invariants at the infinite places. Algebras with ramification at the infinite place seem to provide better diversity-multiplexing gain tradeoff.
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Division Algebra codes achieve MIMO block fading channel capacity within a constant gap
2015Co-Authors: Laura Luzzi, Roope VehkalahtiAbstract:This work addresses the question of achieving capacity with lattice codes in multi-antenna block fading channels when the number of fading blocks tends to infinity. In contrast to the standard approach in the literature which employs random lattice ensembles, the existence results in this paper are derived from number theory. It is shown that a multiblock construction based on Division Algebras achieves rates within a constant gap from block fading capacity both under maximum likelihood decoding and naive lattice decoding. First the gap to capacity is shown to depend on the discriminant of the chosen Division Algebra; then class field theory is applied to build families of Algebras with small discriminants. The key element in the construction is the choice of a sequence of Division Algebras whose centers are number fields with small root discriminants.
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A new design criterion for spherically-shaped Division Algebra-based space-time codes
2013Co-Authors: Laura Luzzi, Roope VehkalahtiAbstract:This work considers normalized inverse determinant sums as a tool for analyzing the performance of Division Algebra based space-time codes for multiple antenna wireless systems. A general union bound based code design criterion is obtained as a main result. In our previous work, the behavior of inverse determinant sums was analyzed using point counting techniques for Lie groups; it was shown that the asymptotic growth exponents of these sums correctly describe the diversity-multiplexing gain trade-off of the space-time code for some multiplexing gain ranges. This paper focuses on the constant terms of the inverse determinant sums, which capture the coding gain behavior. Pursuing the Lie group approach, a tighter asymptotic bound is derived, allowing to compute the constant terms for several classes of space-time codes appearing in the literature. The resulting design criterion suggests that the performance of Division Algebra based codes depends on several fundamental Algebraic invariants of the underlying Algebra.
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connecting dmt of Division Algebra space time codes and point counting in lie groups
International Symposium on Information Theory, 2012Co-Authors: Roope Vehkalahti, Laura LuzziAbstract:Earlier it was proven by Vehkalahti and Lu how the unit group and diversity-multiplexing gain trade-off (DMT) of Division Algebra-based space-time codes are linked to each other through inverse determinant sums. This work explores this relation further, showing that indeed the density of unit group completely determines the growth of the inverse determinant sum. In particular, in the case of Q(i)-central Division Algebras, the lower bound obtained from the DMT and the upper bound derived from the growth rate of units coincide.
Adrian R. Wadsworth - One of the best experts on this subject based on the ideXlab platform.
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Galois subfields of tame Division Algebras
Israel Journal of Mathematics, 2016Co-Authors: Timo Hanke, Danny Neftin, Adrian R. WadsworthAbstract:We show that a finite-dimensional tame Division Algebra D over a Henselian field F has a maximal subfield Galois over F if and only if its residue Division Algebra \(\overline D \) has a maximal subfield Galois over the residue field \(\overline F \).
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Indecomposable Division Algebras
Springer Monographs in Mathematics, 2015Co-Authors: Jean-pierre Tignol, Adrian R. WadsworthAbstract:A central Division Algebra D over a field F is said to be decomposable if D=D 1⊗ F D 2 for some proper subAlgebras D 1, D 2 of D; otherwise, it is indecomposable. In this chapter we give examples of indecomposable Algebras, emphasizing constructions that use valuation theory. In light of the primary decomposition, we restrict attention to Division Algebras of prime power degree. Indecomposable Algebras of exponent p 2 or higher are relatively easy to construct as “p-th roots” of other Division Algebras. Such constructions are discussed in §10.1, where we also give an example of an indecomposable Division Algebra D that becomes decomposable after a scalar extension of degree prime to \(\operatorname {\mathit{deg}}D\). §10.2 focuses on the more difficult case of indecomposables of prime exponent. We give in §10.2.1 a criterion of Jacob to test the decomposability of a tame semiramified Division Algebra of prime exponent over a Henselian field. This criterion yields examples of exponent 2 and degree 8 in §10.2.2, and of exponent p≠2 and degree p r with r≥2 in §10.2.4. The last section, §10.3, deals with complete decompositions into tensor products of symbol Algebras. The main result is Th. 10.26, which relates armatures in an inertially split Division Algebra over a Henselian field to armatures in special representatives of its specialization coset.
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Galois subfields of tame Division Algebras
arXiv: Rings and Algebras, 2013Co-Authors: Timo Hanke, Danny Neftin, Adrian R. WadsworthAbstract:We show that a finite-dimensional tame Division Algebra D over a Henselian field F has a maximal subfield Galois over F if and only if its residue Division Algebra has a maximal subfield Galois over the residue field of F. This generalizes the mechanism behind several known noncrossed product constructions to a crossed product criterion for all tame Division Algebras, and in particular for all Division Algebras if the residue characteristic is 0. If the residue field is a global field, the criterion leads to a description of the location of noncrossed products among tame Division Algebras, and their discovery in new parts of the Brauer group.
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SK1 of graded Division Algebras
Israel Journal of Mathematics, 2011Co-Authors: Roozbeh Hazrat, Adrian R. WadsworthAbstract:The reduced Whitehead group SK1 of a graded Division Algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that SK1 of a tame valued Division Algebra over a henselian field coincides with SK1 of its associated graded Division Algebra. Furthermore, it is shown that SK1 of a graded Division Algebra is isomorphic to SK1 of its quotient Division Algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued Division Algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes SK1 for generic abelian crossed products.
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On maximal Subgroups of the multiplicative group of a Division Algebra
arXiv: Rings and Algebras, 2008Co-Authors: Roozbeh Hazrat, Adrian R. WadsworthAbstract:The question of existence of a maximal subgroup in the multiplicative group D* of a Division Algebra D finite dimensional over its center F is investigated. We prove that if D* has no maximal subgroup, then deg(D) is not a power of 2, F^{*2} is divisible, and for each odd prime p dividing deg(D), there exist noncyclic Division Algebras of degree p over F.
Ernst Dieterich - One of the best experts on this subject based on the ideXlab platform.
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the double sign of a real Division Algebra of finite dimension greater than one
Mathematische Nachrichten, 2012Co-Authors: Erik Darpo, Ernst DieterichAbstract:For any real Division Algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a is an element of A\{0} are shown ...
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the double sign of a real Division Algebra of finite dimension greater than one
arXiv: Rings and Algebras, 2011Co-Authors: Erik Darpo, Ernst DieterichAbstract:For any real Division Algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by a non-zero element are shown to form an invariant of A, called its double sign. The double sign causes the category of all real Division Algebras of a fixed dimension n>1 to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of the category of all finite-dimensional real Division Algebras.
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the degree of an eight dimensional real quadratic Division Algebra is 1 3 or 5
Bulletin Des Sciences Mathematiques, 2010Co-Authors: Ernst Dieterich, Ryszard L RubinszteinAbstract:A celebrated theorem of Hopf (1940) [11], Bott and Milnor (1958) [1], and Kervaire (1958) [12] states that every finite-dimensional real Division Algebra has dimension 1, 2, 4, or 8. While the real Division Algebras of dimension 1 or 2 and the real quadratic Division Algebras of dimension 4 have been classified (Dieterich (2005) [6], Dieterich (1998) [3], Dieterich and Ohman (2002) [9]), the problem of classifying all 8-dimensional real quadratic Division Algebras is still open. We contribute to a solution of that problem by proving that every 8-dimensional real quadratic Division Algebra has degree 1, 3, or 5. This statement is sharp. It was conjectured in Dieterich et al. (2006) [7].
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the degree of an eight dimensional real quadratic Division Algebra is 1 3 or 5
arXiv: Rings and Algebras, 2009Co-Authors: Ernst Dieterich, Ryszard L RubinszteinAbstract:A celebrated theorem of Hopf, Bott, Milnor, and Kervaire states that every finite-dimensional real Division Algebra has dimension 1, 2, 4, or 8. While the real Division Algebras of dimension 1 or 2 and the real quadratic Division Algebras of dimension 4 have been classified, the problem of classifying all 8-dimensional real quadratic Division Algebras is still open. We contribute to a solution of that problem by proving that every 8-dimensional real quadratic Division Algebra has degree 1, 3, or 5. This statement is sharp.
Igor A Rapinchuk - One of the best experts on this subject based on the ideXlab platform.
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The finiteness of the genus of a finite-dimensional Division Algebra, and some generalizations
arXiv: Rings and Algebras, 2018Co-Authors: Vladimir Chernousov, Andrei S Rapinchuk, Igor A RapinchukAbstract:We prove that the genus of a finite-dimensional Division Algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the genus of simple Algebraic groups of type $\textsf{G}_2$. These applications involve the double cosets of adele groups of Algebraic groups over arbitrary finitely generated fields: while over number fields these double cosets are associated with the class numbers of Algebraic groups and hence have been actively analyzed, similar question over more general fields seem to come up for the first time. In the Appendix, we link the double cosets with $\check{\rm C}$ech cohomology and indicate connections between certain finiteness properties involving double cosets (Condition (T)) and Bass's finiteness conjecture in $K$-theory.
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On the size of the genus of a Division Algebra
Proceedings of the Steklov Institute of Mathematics, 2016Co-Authors: Vladimir Chernousov, Andrei S Rapinchuk, Igor A RapinchukAbstract:Let D be a central Division Algebra of degree n over a field K. One defines the genus gen(D) as the set of classes [D′] ∈ Br(K) in the Brauer group of K represented by central Division Algebras D′ of degree n over K having the same maximal subfields as D. We prove that if the field K is finitely generated and n is prime to its characteristic, then gen(D) is finite, and give explicit estimations of its size in certain situations.
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The genus of a Division Algebra and the unramified Brauer group
Bulletin of Mathematical Sciences, 2013Co-Authors: Vladimir Chernousov, Andrei S Rapinchuk, Igor A RapinchukAbstract:Let $$D$$ be a finite-dimensional central Division Algebra over a field $$K$$ . We define the genus $$\mathbf{gen}(D)$$ of $$D$$ to be the collection of classes $$[D^{\prime }] \in \mathrm{Br}(K)$$ , where $$D^{\prime }$$ is a central Division $$K$$ -Algebra having the same maximal subfields as $$D$$ . In this paper, we describe a general approach to proving the finiteness of $$\mathbf{gen}(D)$$ and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of $$K$$ . This approach is then implemented in some concrete situations, yielding in particular an extension of the Stability Theorem of A. Rapinchuk and I. Rapinchuk (Manuscr. Math. 132:273–293, 2010 ) from quaternion Algebras to arbitrary Algebras of exponent two. We also consider an example where the size of the genus can be estimated explicitly. Finally, we offer two generalizations of the genus problem for Division Algebras: one deals with absolutely almost simple Algebraic $$K$$ -groups having the same isomorphism/isogeny classes of maximal $$K$$ -tori, and the other with the analysis of weakly commensurable Zariski-dense subgroups.
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the genus of a Division Algebra and the unramified brauer group
arXiv: Rings and Algebras, 2013Co-Authors: Vladimir Chernousov, Andrei S Rapinchuk, Igor A RapinchukAbstract:Let D be a finite-dimensional central Division Algebra over a field K. We define the genus gen(D) of D to be the collection of classes in the Brauer group of K represented by central Division K-Algebras D' having the same maximal subfields as D. In this paper, we describe a general approach to proving the finiteness of gen(D) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of K.
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on the genus of a Division Algebra
Comptes Rendus Mathematique, 2012Co-Authors: Vladimir Chernousov, Andrei S Rapinchuk, Igor A RapinchukAbstract:Abstract We define the genus gen ( D ) of a finite-dimensional central Division Algebra D over a field K as the set of all classes [ D ′ ] in the Brauer group Br ( K ) that are represented by central Division K-Algebras D ′ having the same maximal subfields as D. We give examples where gen ( D ) is reduced to a single element, and other examples where it is finite.