A New Tool: Constructing STBCs from Maximal Orders in Central Simple Algebras

Hollanti, Camilla; Lahtonen, Jyrki

doi:10.1109/itw.2006.1633838

Cited by 15 publications

(15 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In view of all of the above this is also a consequence of the fact that the Z[i]-discriminant of the natural order of the Golden algebra is equal to 25. As was observed in [25] the natural order happens to be maximal in this case, so the Golden code cannot be improved upon by enlarging the order within GA.…”

Section: Example 24mentioning

confidence: 76%

See 1 more Smart Citation

On the Densest MIMO Lattices From Cyclic Division Algebras

Vehkalahti

Hollanti

Lahtonen

et al. 2009

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

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

show abstract

Section: Example 24mentioning

confidence: 76%

“…We also suppose that p is inert in the extension F/Q. 25 The extension F Z/F is a cyclic Galois extension of degree q n where the prime ideals P 1 and P 2 are totally inert and P = pO F is the only ramified prime ideal in the extension F Z/F .…”

Section: B General Constructionmentioning

confidence: 99%

On the Densest MIMO Lattices From Cyclic Division Algebras

Vehkalahti

Hollanti

Lahtonen

et al. 2009

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

show abstract

“…In view of all of the above this is also a consequence of the fact that the Z[i]-discriminant of the natural order of the Golden algebra is equal to 25. As was observed in [1] the natural order happens to be maximal in this case, so the Golden code cannot be improved upon by enlarging the order within GA.…”

Section: Definition 22mentioning

confidence: 76%

“…It is widely known how the so called Alamouti design represents multiplication in the ring of quaternions. Matrix representations of other division algebras have been proposed as STBC codes at least in [1]- [5], and (though without explicitly saying so) [6]. The most recent work ([2]- [6]) has concentrated on adding multiplexing gain (i.e.…”

Section: Introductionmentioning

confidence: 99%

Optimal Matrix Lattices for MIMO Codes from Division Algebras

Hollanti¹,

Lahtonen

Ranto

et al. 2006

2006 IEEE International Symposium on Information Theory

View full text Add to dashboard Cite

We show 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 minimal determinant. Using results from class field theory we derive a lower bound to the minimum discriminant of a maximal order with a given center and index (= the number of Tx/Rx antennas). We also give examples of division algebras achieving our bound. E.g. we construct a matrix lattice with QAM coefficients that has (inside 'large' subsets of the signal space) 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant. We also give another matrix lattice with coefficients from the hexagonal lattice with an even higher density.

show abstract

“…Let C be a Fuchsian code attached to a group Γ containing a nonabelian free subgroup. Then the complexity can be upper bounded as r C ≤ r C = κ 0 (5M +14) log(|C| + 2) log(2) − 2 + 5M + 7,…”

mentioning

confidence: 99%

Nonuniform Fuchsian codes for noisy channels

Blanco-Chacón

Remón

Hollanti

et al. 2014

Journal of the Franklin Institute

View full text Add to dashboard Cite

We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not apply. Previously, only brute force decoding methods with complexity that is linear in the code size exist for general nonuniform codes. However, the properly discontinuous character of the action of the Fuchsian groups on the complex upper half-plane translates into decoding complexity that is logarithmic in the code size via a recently introduced point reduction algorithm.Peer ReviewedPostprint (author’s final draft

show abstract

A New Tool: Constructing STBCs from Maximal Orders in Central Simple Algebras

Cited by 15 publications

References 11 publications

On the Densest MIMO Lattices From Cyclic Division Algebras

On the Densest MIMO Lattices From Cyclic Division Algebras

Optimal Matrix Lattices for MIMO Codes from Division Algebras

Nonuniform Fuchsian codes for noisy channels

Contact Info

Product

Resources

About