Space-Time Block Codes from Cyclic Division Algebras: An Introduction

Coding theory addresses the problem of transmitting information accurately across noisy channels. When a sender transmits a signal s, it will suffer some changes before it reaches the receiver. The receiver then faces the problem of recovering the intended signal, given that he actually received some altered signal r. The challenge of coding theory, then, is to design a system which not only gives the receiver a high probability of determining s given r, but is also fast and inexpensive. Most solutions to this problem involve choosing, among all the signals that could be transmitted, some subset of “legal” codewords which are not too similar to one another. Then if r is in fact a codeword, the receiver may assume that r = s; otherwise, presumably s was a codeword similar to r. Natural languages incorporate such “redundancy” already; if written words are misspelled, the reader can very often reconstruct the original text of the author verbatim. Devising some analogue of this natural process for arbitrary data is the task of coding theory. Historically, the codewords used were vectors, but in recent years codes using matrices have been developed. In early 1998 Tarokh, Seshadri, and Calderbank proposed a system called “space-time coding” [7], and later that year the classical example, the Alamouti scheme, was published [1]. Efforts to generalize Alamouti have recently turned to division algebras to obtain good collections of matrices for codebooks. This paper will first briefly introduce the basic concepts needed to understand spacetime block codes, outlining the criteria a good code must satisfy; it then will summarize the constructions used in two recent papers to produce workable space-time block codes from cyclic division algebras.