Cyclic Division Algebras: A Tool for Space-Time Coding

Abstract

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.

Layered Space-Time Index Coding

“…Following [22], we define A (X − X ′ )(X − X ′ ) † for any pair of codeword matrices X, X ′ ∈ C. Let r be the rank of A. For the generic user with S = ∅, in the high SNR regime, the probability of mistaking X ′ for X can be bounded as…”

“…A cyclic division algebra is a cyclic algebra that is at the same time a division algebra. In the space-time coding literature (see [22] and reference therein), a cyclic division algebra is usually constructed from a cyclic algebra A = (L/K, σ, γ) with carefully chosen γ such that none of γ, γ 2 , . .…”

“…The multicasting problem with message side information is then considered in [18] under the 2×2 MIMO setting where the transmitter and the receivers are equipped with two antennas. For such a MIMO setting, the minimum determinant of the code serves as one of the most important parameters to be maximized [21], [22] and algebraic space-time block codes (STBC) constructed from cyclic division algebras [23]- [26] are a class of codes that possess many desired properties. Since CRT does not hold for non-commutative rings such as cyclic division algebras, the trick used in [16] and [17] does not work here in general.…”

“…To make the paper concise, we only review the minimum required background for understanding the discussion that follows. For details, please refer, for example, to [20], [22], [29]- [31].…”

Golden-coded index coding

“…We then study the construction of good space-time index codes. While there is a rich literature in the study of construction of space-time codes for the point-to-point MIMO channel (see [7] and the reference therein), as a first attempt, we consider construction of lattice space-time index codes solely based on golden codes [8] for the 2 × 2 case. The main difficulty is that most of the code constructions proposed in [5] and [6] rely on partitions induced by the Chinese remainder theorem (CRT) for some commutative rings; however, golden codes (and most of the lattice space-time codes) are constructed over a cyclic division algebra, which is noncommutative and hence prevents the direct application of CRT.…”