Maximal Orders in the Design of Dense Space-Time Lattice Codes

Hollanti, Camilla; Lahtonen, Jyrki; Lu, Hsiao-feng

doi:10.1109/tit.2008.928998

Cited by 44 publications

(31 citation statements)

References 52 publications

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“…In [5] we did the same but for the purpose of saving energy and making the lattice easier to encode. We include the following simple fact (also known to E. Viterbo, private communication) explaining why using a principal one-sided (left or right) ideal instead of the entire order will not change the density of the code.…”

Section: An Example Code and Some Simulation Resultsmentioning

confidence: 99%

“…Example 6.4: From Table III we are division algebras with minimal discriminants. According to Proposition 5.1 algebra A 2 ⊗ A 3 = (Q(i)(a 6 )/Q(i), σ 2 σ 3 , (2 + i) 5 (1 + i) 2 ), where a 6 is a zero of the polynomial x 6 − 2x 5 + (−3i − 51)x 4 + (4i − 30)x 3 + (−2i + 755)x 2 + (−298i + 2134)x + −593i + 1628, is a division algebra of degree 6 and has a minimal discriminant.…”

Section: Existence Of Suitable Primesmentioning

confidence: 99%

“…the resulting STBC meets the rank criterion. Matrix representations of other division algebras have been proposed as STBCs at least in [5]- [14], and (though without explicitly saying so) [15]. The most recent work [7]- [15] has concentrated on adding multiplexing gain, i.e.…”

mentioning

confidence: 99%

“…For more details about this see Section VII below. Earlier we have successfully used maximal orders in a construction of some 4Tx antenna MISO lattices [5].…”

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confidence: 99%

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On the Densest MIMO Lattices From Cyclic Division Algebras

Vehkalahti

Hollanti

Lahtonen

et al. 2009

IEEE Trans. Inform. Theory

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It is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant. Using results from class field theory a lower bound to the minimum discriminant of a maximal order with a given center and index (= the number of Tx/Rx antennas) is derived. Also numerous examples of division algebras achieving our bound are given. E.g. we construct a matrix lattice with QAM coefficients that has 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant. We describe a general algorithm due to Ivanyos and Rónyai for finding maximal orders within a cyclic division algebra and discuss our enhancements to this algorithm. We also consider general methods for finding cyclic division algebras of a prescribed index achieving our lower bound. Index TermsCyclic division algebras, dense lattices, discriminants, Hasse invariants, maximal orders, multiple-input multipleoutput (MIMO) channels, multiplexing, space-time block codes (STBCs).I. OVERVIEW Multiple-antenna wireless communication promises very high data rates, in particular when we have perfect channel state information (CSI) available at the receiver. In [1] the design criteria for such systems were developed, and further on the evolution of space-time (ST) codes took two directions: trellis codes and block codes. Our work concentrates on the latter branch.We are interested in the coherent multiple input-multiple output (MIMO) case. A lattice is a discrete finitely generated free abelian subgroup L of a real or complex finite dimensional vector space V, called the ambient space. In the space-time setting a natural ambient space is the space M n (C) of complex n×n matrices. We only consider full rank lattices that have a basis x 1 , x 2 , . . . , x 2n 2 consisting of matrices that are linearly independent over the field of real numbers. We can form a 2n 2 × 2n 2 matrix M having rows consisting of the real and imaginary parts of all the basis elements. It is well known that the measure, or hypervolume, m(L) of the fundamental parallelotope of the lattice then equals the absolute value of det(M). Alternatively we may use the Gram matrix 2 . From the pairwise error probability (PEP) point of view [2], the performance of a space-time code is dependent on two parameters: diversity gain and coding gain. Diversity gain is the minimum of the rank of the difference matrix X − X ′ taken over all distinct code matrices X, X ′ ∈ C, also called the rank of the code C. When C is full-rank, the coding gain is proportional to the determinant of the matrix

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Section: An Example Code and Some Simulation Resultsmentioning

confidence: 99%

Section: Existence Of Suitable Primesmentioning

confidence: 99%

mentioning

confidence: 99%

“…For more details about this see Section VII below. Earlier we have successfully used maximal orders in a construction of some 4Tx antenna MISO lattices [5].…”

mentioning

confidence: 99%

See 2 more Smart Citations

On the Densest MIMO Lattices From Cyclic Division Algebras

Vehkalahti

Hollanti

Lahtonen

et al. 2009

IEEE Trans. Inform. Theory

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“…One of the most promising extensions is by using maximal orders of the algebra in order to have a larger set of codewords with at least 6.4 Conclusion 89 the same minimum determinant [30]. There are also other applications for division algebras based codes than the MIMO or the Relay channel such as, for instance, the MIMO-ARQ channel [47].…”

Section: Other Issuesmentioning

confidence: 99%

Cyclic Division Algebras: A Tool for Space-Time Coding

Oggier

Belfiore

Viterbo

2007

FNT in Communications and Information Theory

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Multiple antennas at both the transmitter and receiver ends of a wireless digital transmission channel may increase both data rate and reliability. Reliable high rate transmission over such channels can only be achieved through Space-Time coding. Rank and determinant code design criteria have been proposed to enhance diversity and coding gain. The special case of full-diversity criterion requires that the difference of any two distinct codewords has full rank.Extensive work has been done on Space-Time coding, aiming at finding fully diverse codes with high rate. Division algebras have been proposed as a new tool for constructing Space-Time codes, since they are non-commutative algebras that naturally yield linear fully diverse codes. Their algebraic properties can thus be further exploited to improve the design of good codes.The aim of this work is to provide a tutorial introduction to the algebraic tools involved in the design of codes based on cyclic division algebras. The different design criteria involved will be illustrated, including the constellation shaping, the information lossless property, the non-vanishing determinant property, and the diversity multiplexing trade-off. The final target is to give the complete mathematical background underlying the construction of the Golden code and the other Perfect Space-Time block codes.

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Natural orders for asymmetric space–time coding: minimizing the discriminant

Barreal

Rodrigáñez

Hollanti

2017

AAECC

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Algebraic space-time coding -a powerful technique developed in the context of multiple-input multiple-output (MIMO) wireless communications -has profited tremendously from tools from Class Field Theory and, more concretely, the theory of central simple algebras and their orders. During the last decade, the study of space-time codes for practical applications, and more recently for future generation (5G+) wireless systems, has provided a practical motivation for the consideration of many interesting mathematical problems. One such problem is the explicit computation of orders of central simple algebras with small discriminants. In this article, we consider the most interesting asymmetric MIMO channel setups and, for each treated case, we provide explicit pairs of fields and a corresponding non-norm element giving rise to a cyclic division algebra whose natural order has the minimum possible discriminant.

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Maximal Orders in the Design of Dense Space-Time Lattice Codes

Cited by 44 publications

References 52 publications

On the Densest MIMO Lattices From Cyclic Division Algebras

On the Densest MIMO Lattices From Cyclic Division Algebras

Cyclic Division Algebras: A Tool for Space-Time Coding

Natural orders for asymmetric space–time coding: minimizing the discriminant

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