## Introduction

Space-time coding refers to coding for wireless systems
equipped with several antennas at both the transmitter and receiver. Codewords are
matrices with complex coefficients, and the main design criterion is called the
rank criterion. For square matrices, it says that for any pair of matrices in the code, the determinant of their difference must be non-zero, to ensure full diversity, which translates into good performance of the code. This project studies different aspects of space-time coding.

## Fast Decodable Space-Time Codes

The bottleneck for using many space-time codes is their
decoding complexity. Fast decodable space-time codes refer to codes which are decoded using a Sphere Decoder algorithm (which essentially performs a closest lattice point search) whose decoding complexity is reduced with respect to doing an exhaustive search.
We have so far studied asymmetric systems where the transmitter has more antennas than the receiver, and in particular the case of 4 antennas at the transmitter and 2 at the receiver, for which the complexity of exhaustive search is O(M

In

By systematically studying crossed product algebras over Q, we found in

In

^{16}) if M denotes the size of the real signal alphabet.In

**[C4]**we showed that by puncturing codes from crossed product algebras, we can obtain codes with reduced complexity and the so-called non-vanishing determinant property, which guarantees a given coding gain independently of the signal constellation size. We proposed other codes with similar properties in**[C5]**, while a further puncturing revealed that a MISO code could further be embedded in a MIDO code. We achieved so far a complexity of O(M^{12}). A general theory that gives criteria to embed division algebras into matrices with quaternionic coefficients, as well as several further code constructions, including for 6 transmit antennas, is presented in**[J3]**.By systematically studying crossed product algebras over Q, we found in

**[C6]**different constructions with complexity O(M^{10}).In

**[C7]**, we propose a different way to build algebraic fast decodable codes using an iterated construction, which from an algebraic 2x2 space-time code coming from a quaternion algebra derives a 4x4 space-time code, similarly to the way the quasi-orthogonal space-time code proposed by Jafarkhani was obtained from the Alamouti block code. The case when the original space-time code is a 3x3 code is treated in**[C9]**. This method is general in that it works with for an nxn space-time code coming from a cyclic algebra**[J4]**, crossed product algebras**[C8]**, and can be iterated several times**[C10]**. Some fast decodable constructions from non-associative algebras have been investigated in**[C12]**.## Space-Time Coded Modulation

This project considers coding for slow fading MIMO channels.
In this case a codeword can be seen as a sequence of L square matrices. If each square matrix is fully diverse and encoded independently, good performance is obtained, however it can be improved by jointly encoding the L fully diverse matrices. In this work, we consider coset coding, and show that if we start by space-time codes coming from division algebras, designing coset codings translates to design codes over matrices over finite rings. The case of 2x2 matrices with the Golden code as inner code is studied first in

**[C2]**, and the general framework to consider perfect codes in higher dimension is presented in**[C3]**. Code constructions for perfect codes in dimension 2,3, and 4 are given in**[J2]**. A general framework to study nxn space-time codes coming from division algebras is studied in**[J5]**.## Publications

- Journals
**[J5]**F. Oggier, B. A. Sethuraman. Quotients of Orders in Cyclic Algebras and Space-Time Codes,*Advances in Mathematics of Communications*, vol. 7, no. 4, 2013.**[J4]**N. Markin, F. Oggier. Iterated Space-Time Code Constructions from Cyclic Algebras,*IEEE Transactions on Information Theory*, no. 9, vol. 59, September 2013.**[J3]**R. Vehkalahti, C. Hollanti, F. Oggier. Fast-Decodable Asymmetric Space-Time Codes from Division Algebras,*IEEE Transactions on Information Theory*, Vol. 58, No. 4, April 2012.**[J2]**F. Oggier, P. Solé, J.-C. Belfiore. Codes over Matrix Rings for Space-Time Coded Modulation.*IEEE Transactions on Information Theory*, Vol. 58, No.2, February 2012.**[J1]**F. Oggier, E. Lequeu. Differential Distributed Cayley Space-Time Codes,*IEEE Transactions in Wireless Communications*, Vol.8, No.7, July 2009.- Conferences
**[C13]**C. Hollanti, N. Markin, A unified framework for constructing fast-decodable codes for N relays.*Extended abstract, MTNS 2012.***[C12]**A. Steele, S. Pumpluen, F. Oggier, MIDO Space-Time Codes from Associative and Nonassociative Cyclic Algebras,*ITW 2012*.**[C11]**C. Hollanti, N. Markin, Algebraic Fast-Decodable Relay Codes for Distributed Communications,*ISIT 2012*.**[C10]**N. Markin, F. Oggier, A Class of Iterated Fast Decodable Space-Time Codes for 2^{n}Tx Antennas,*ISIT 2012*.**[C9]**N. Markin, F. Oggier, Fast Decodable Codes for 6Tx-3Rx MIMO Systems,*SPCOM 2012.***[C8]**N. Markin, F. Oggier, Iterated Fast Decodable Space-Time Codes From Crossed-Products.*Extended abstract, MTNS 2012.***[C7]**N. Markin, F. Oggier. Iterated MIDO Space-Time Code Constructions .*Allerton 2011*.**[C6]**L. Luzzi, F. Oggier. A family of fast-decodable MIDO codes from crossed-product algebras over Q.*ISIT 2011*.**[C5]**F. Oggier, C. Hollanti, R. Vehkalahti. An Algebraic MIDO-MISO Code Construction.*SPCOM 2010*.**[C4]**F. Oggier, R. Vehkalahti, C. Hollanti. Fast Decodable MIDO Codes from Crossed Product Algebras.*ISIT 2010*.**[C3]**F. Oggier, P. Solé. Higher Dimensional Perfect Space-Time Coded Modulation.*ITW 2009*.*Best poster award*.**[C2]**F. Oggier, P. Solé, J.-C. Belfiore. Codes over M2(F2) and Applications to Golden Space-Time Modulation .*ISIT 2009*.**[C1]**F. Oggier. A Survey of Algebraic Unitary Codes .*IWCC 09*.- Book
- Invited Talks
- On Algebraic Space-Time Codes , by N. Markin, ECE Department, IISC Bangalore, August 1 2012.
- On Space-Time Codes Arising from Algebras, by N. Markin, Department of Mathematics, University of Turku, January 18 2012.
- On Space-Time Codes Arising from Algebras, by N. Markin, Coordinated Science Laboratory, UIUC, USA, October 7 2011.
- Algèbres centrales simples pour le codage espace-temps, plenary talk by F. Oggier, French Mathematical Society Annual Meeting, Paris, France, June 26 2010.
- Ideal Lattices and Codes, mini-course by F. Oggier, GTEM Summer School, EPFL, Lausanne, Switzerland, July 2009.
- Introduction to Algebraic Wireless Coding, by F. Oggier, University of Calabria, Italy, May 14 2009.