Communication over a wireless channel is a challenging task because of the inherent fading effects. Any wireless communication system employs some form of diversity improving techniques in order to improve the reliability of the channel. This thesis deals with efficient code design for two different spatial diversity techniques, viz, diversity by employing multiple antennas at the transmitter and/or the receiver, and diversity through cooperative commu- nication between users. In other words, this thesis deals with efficient code design for (1) multiple-input multiple-output (MIMO) channels, and (2) wireless relay channels. Codes for the MIMO channel are termed space-time (ST) codes and those for the relay channels are called distributed ST codes. The first part of the thesis focuses on ST code construction for MIMO fading channel with perfect channel state information (CSI) at the receiver, and no CSI at the transmitter. As a measure of performance we use the rate-diversity tradeoff and the Diversity-Multiplexing Gain (D-MG) Tradeoff, which are two different tradeoffs characterizing the tradeoff between the rate and the reliability achievable by any ST code. We provide two types of code constructions that are optimal with respect to the rate-diversity tradeoff; one is based on the rank-distance codes which are traditionally applied as codes for storage devices, and the second construction is based on a matrix representation of a cayley algebra. The second contribution in ST code constructions is related to codes with a certain nonvanishing determinant (NVD) property. Motivation for these constructions is a recent result on the necessary and sufficient conditions for an ST code to achieve the D-MG tradeoff. Explicit code constructions satisfying these conditions are provided for certain number of transmit antennas. The second part of the thesis focuses on distributed ST code construction for wireless relay channel. The transmission protocol follows a two-hop model wherein the source broadcasts a vector in the first hop and in the second hop the relays transmit a vector that is a transformation of the received vector by a relay-specific unitary transformation. While the source and relays do not have CSI, at the destination we assume two different scenarios (a) destina- tion with complete CSI (b) destination with only the relay-destination CSI. For both these scenarios, we derive a Chernoff bound on the pair-wise error probability and propose code design criteria. For the first case, we provide explicit construction of distributed ST codes with lower decoding complexity compared to codes based on some earlier system models. For the latter case, we propose a novel differential encoding and differential decoding technique and also provide explicit code constructions. At the heart of all these constructions is the cyclic division algebra (CDA) and its matrix representations. We translate the problem of code construction in each of the above scenarios to the problem of constructing CDAs satisfying certain properties. Explicit examples are provided to illustrate each of these constructions.

This thesis addresses the design of low-complexity coding schemes for wireless relay networks and multiple access channels. The first part of the thesis is on wireless relay networks and the second part is on multiple access channels. Distributed space-time coding is a well known technique to achieve spatial diversity in wireless networks wherein, several geographically separated nodes assist a source node to distributively transmit a space-time block code (STBC) to the destination. Such STBCs are referred to as Distributed STBCs (DSTBCs). In the first part of the thesis, we focus on designing full diversity DSTBCs with some nice properties which make them amenable for implementation in practice. Towards that end, a class of full diversity DST-BCs referred to as Co-ordinate Interleaved DSTBCs (CIDSTBCs) are proposed for relay networks with two-antenna relays. To construct CIDSTBCs, a technique called co-ordinate vector interleaving is introduced wherein, the received signals at different antennas of the relay are processed in a combined fashion. Compared to the schemes where the received signals at different antennas of the relay are processed independently, we show that CIDSTBCs provide coding gain which comes in with negligible increase in the processing complexity at the relays. Subsequently, we design single-symbol ML decodable (SSD) DSTBCs for relay networks with single-antenna nodes. In particular, two classes of SSD DSTBCs referred to as (i) Semi-orthogonal SSD Precoded DSTBCs and (ii) Training-Symbol Embedded (TSE) SSD DSTBCs are proposed. A detailed analysis on the maximal rate of such DSTBCs is presented and explicit DSTBCs achieving the maximal rate are proposed. It is shown that the proposed codes have higher rates than the existing SSD DSTBCs. In the second part, we study two-user Gaussian Multiple Access Channels (GMAC). Capacity regions of two-user GMAC are well known. Though, capacity regions of such channels provide insights into the achievable rate pairs in an information theoretic sense, they fail to provide information on the achievable rate pairs when we consider finitary restrictions on the input alphabets and analyze some real world practical signal constellations like QAM and PSK signal sets. Hence, we study the capacity aspects of two-user GMAC with finite input alphabets. In particular, Constellation Constrained (CC) capacity regions of two-user SISO-GMAC are computed for several orthogonal and non-orthogonal multiple access schemes (abbreviated as O-MA and NO-MA schemes respectively). It is first shown that NO-MA schemes strictly offer larger capacity regions than the O-MA schemes for finite input alphabets. Subsequently, for NO-MA schemes, code pairs based on Trellis Coded Modulation (TCM) are proposed such that any rate pair on the CC capacity region can be approached. Finally, we consider a two-user Multiple-Input Multiple-Output (MIMO) fading MAC and design STBC pairs such that ML decoding complexity is reduced.