Abstract Lattice codes achieve AWGN capacity and naturally fit in many multi-terminal networks because of their inherited structure. Although extensive information theoretic research has been done to prove the importance of lattice codes for these networks, the progress in finding practical low-complexity lattice schemes is limited. Hence, the motivation of this thesis is to develop several methods to make lattice codes practical for communication networks. First, we propose an efficient lattice coding scheme for real-valued, full-duplex one- and two-way relay channels. Lattice decomposition, superposition, and block Markov encoding are used to propose a simple, yet near capacity achieving encoding/decoding schemes for these relay channels. By using information theoretic tools, we prove the achievable rates of these schemes, which are equal to the best known rates. Then, we construct practical, low-complexity implementations of the proposed relay schemes using low-density lattice codes. Numerical evaluation is presented and they show that our schemes achieve performance as close as 2.5dB away from theoretical limits. The effect of shaping/coding loss on the performance of relay channels is studied. Then, we propose a low complexity lattice code construction that provides high shaping and coding gains. First, integer information is encoded to shaped integers. Two methods are proposed for this task: ''Voronoi integers'' and ''non uniform integers''. These shaped integers have shaping gains over the integer lattice. Then for the second step, we present a general framework to systematically encode these integers, using any high dimensional lattice with lower-triangular generator or parity check matrices, retaining the same shaping gain. The proposed scheme can be used to shape high dimensional lattices such as low density lattice codes, LDA-lattice, etc. Comprehensive analysis is presented using low density lattice codes. By using E8 and BW16 as shaping lattices, we numerically show the Voronoi integers result in the shaping gain of these lattices, that is, as much as 0.65dB and 0.86dB. It is numerically observed that non-uniform integers have shaping gains of up to 1.25dB. These shaping operations can be implemented with less complexity than previous low density lattice codes shaping approaches and shaping gains are higher than in previously reported cases, which are in the order of 0.4dB. Lastly, we propose a low complexity practical code construction for compute-and-forward. A novel code construction called ''mixed nested lattice code construction'' is developed. This code construction uses a pair of distinct nested lattices to encode the integers where shaping is provided by a small dimensional lattice with high shaping gain and coding is performed using a high coding gain and a high dimensional lattice. This construction keeps the shaping and the coding gains of respective shaping and coding lattices. Further, we prove an existence of an isomorphism in this construction such that linear combination of lattice codes can be mapped to a linear combination of integers over a finite field. Hence, this construction can be readily used for any compute-and-forward applications. A modified LDLC decoder is proposed to estimate a linear combination of messages. Performance is numerically evaluated. / Tiivistelmä Hilakoodit saavuttavat AWGN kapasiteetin ja sopivat luonnollisesti moniin monen päätelaitteen verkkoihin niihin sisältyvän rakenteen vuoksi. Vaikka lukuisat informaatioteoreettiset tutkimustyöt todistavat hilakoodien tärkeyden näille verkoille, käytännössä alhaisen kompleksisuuden hilajärjestelmiä on vielä vähän. Näin ollen tämän tutkielman tarkoitus on kehittää useita metodeja, jotta hilakoodeista saadaan käytännöllisiä viestintäverkkoihin. Aluksi, ehdotamme tehokkaan hilakoodausjärjestelmän reaaliarvoisille, full duplexisille yksi- ja kaksisuuntaisille välittäjäkanaville. Käytämme hilan hajottamista, superpositiota ja lohko-Markov -koodausta ehdottaessamme yksinkertaiset ja siltikin kapasiteetin saavuttavat koodaus- ja dekoodausjärjestelmät näihin välityskanaviin. Käyttämällä informaatioteoreettisia työkaluja, osoitamme näiden järjestelmien saavutettavat nopeudet, jotka ovat yhtä suuret kuin parhaimmat tunnetut nopeudet. Sitten rakennamme käytännölliset ja alhaisen monimutkaisuuden toteutukset ehdotetuille välitysjärjestelmille käyttäen alhaisen tiheyden hilakoodeja. Esitämme näille järjestelmille numeeriset arvioinnit, jotka näyttävät että nämä toteutukset saavuttavat tehokkuuden, joka on 2.5dB:n päässä teoreettisista rajoista. Tutkimme muotoilu- ja koodaushäviön vaikutusta välityskanavien tehokkuuteen. Sitten, ehdotamme alhaisen monimutkaisuuden hilakoodirakenteen, joka tarjoaa korkean muotoilu- ja koodausvahvistuksen. Ensin, kokonaislukuinformaatio on koodattu muotoiltuihin kokonaislukuihin. Esitämme kaksi metodia tähän tehtävään; 'Voronoi kokonaisluvut' ja 'ei yhtenäiset kokonaisluvut'. Näillä muotoilluilla kokonaisluvuilla on muotoiluvahvistusta kokonaislukuhilalle. Toisena askeleena, esitämme yleiset puitteet systemaattiseen kokonaislukujen koodaukseen käyttäen korkeaulotteisia hiloja alhaisen kolmiogeneraattori- tai pariteettivarmistusmatriiseja, jotka säilyttävät samalla muotoiluvahvistuksen. Ehdotettua järjestelmää voidaan käyttää muotoilemaan korkeaulotteisia hiloja kuten alhaisen tiheyden hilakoodeja, LDA-hiloja, jne. Esitämme kattavan analyysin käyttäen alhaisen tiheyden hilakoodeja. Käyttämällä muotoiluhiloina E8aa ja BW16a, näytämme numeerisesti 'Voronoi kokonaislukujen' käyttämisen seurauksena saavutettavat hilojen muotoiluvahvistukset, jotka ovat jopa 0.65dB ja 0.86dB. Näytämme myös numeerisesti että 'ei yhtenäisillä kokonaisluvuilla' on muotoiluvahvistusta jopa 1.25dB. Nämä muotoiluoperaatiot voidaan toteuttaa alhaisemmalla monimutkaisuudella kuin aikaisemmat 'alhaisen tiheyden hilakoodien muotoilumenetelmät' ja muotoiluvahvistukset ovat suuremmat kuin aikaisemmin raportoidut tapaukset, jotka ovat suuruusluokaltaan 0.4dB. Viimeiseksi, ehdotamme käytännöllisen koodikonstruktion alhaisella monimutkaisuudella 'laske ja lähetä' -menetelmään. Kehitämme uuden koodikonstruktion, jota kutsumme 'sekoitetuksi sisäkkäiseksi hilakoodikonstruktioksi'. Tämä koodikonstruktio käyttää kahta eroteltavissa olevaa sisäkkäistä hilaa koodaamaan kokonaisluvut siellä, missä muotoilu tehdään pienen ulottuvuuden hiloilla korkean muotoiluvahvistuksella ja koodaus toteutetaan käyttäen korkean koodausvahvistuksen omaavaa korkeaulottuvuuksista hilaa. Tämä konstruktio säilyttää muotoilu- ja koodausvahvistukset kullekin muotoilu- ja koodaushilalle. Lisäksi, todistamme isomorfismin olemassaolon tässä konstruktiossa siten, että lineaarisen hilakoodien kombinaatio voidaan kuvata lineaarisena kokonaislukujen kombinaationa äärellisessä kunnassa. Näin ollen tätä konstruktiota voidaan helposti käyttää missä tahansa 'laske ja lähetä' -sovelluksessa. Esitämme muokatun LDLC dekooderin lineaarisen viestikombinaation estimointiin. Arvioimme tehon numeerisesti.

Cooperative wireless communication has received significant attention during recent years due to several reasons. First, since the received power decreases rapidly with distance, the idea of multi-hopping is becoming of particular importance. In multi-hopped communication, the source exploits some intermediate nodes as relays. Then the source sends its message via those relays to the destination. Second, relays can emulate some kind of distributed transmit antennas to form spatial diversity and combat multi-path fading effect of the wireless channel. Parallel Relay Channel is an information theoretical model for a communication system whereby a sender aims to communicate to a receiver with the help of relay nodes. It represents the simplest model for a multi–hop wireless network and a full understanding of the limits of communication over such a channel can potentially shed light on the design of more efficient wireless networks. However, the capacity of the relay channel has been established only for few special cases and little progress has been made toward solving the general case since the early 1980s. In this dissertation, motivated by practical constraints, we study the information theoretical limits of the half-duplex Gaussian Parallel Relay channel , as well as, the transmission strategies for the parallel relay channel with bandwidth mismatch between the first and the second hops. Chapter 2 investigates the problem of communication for a network composed of two half-duplex parallel relays with additive white Gaussian noise (AWGN). There is no direct link between the source and the destination. However, the relays can communicate with each other through the channel between them. Two protocols, i.e., \emph{Simultaneous} and \emph{Successive} relaying, associated with two possible relay scheduling are proposed. The simultaneous relaying protocol is based on \emph{Broadcast-multiaccess with Common Message (BCM)} scheme. For the successive relaying protocol: (i) a \emph{Non-Cooperative} scheme based on the \emph{Dirty Paper Coding (DPC)}, and (ii) a \emph{Cooperative} scheme based on the \emph{Block Markov Encoding (BME)} are considered. The composite scheme of employing BME in \emph{at most} one relay and DPC in \emph{at least} another one is shown to achieve at least the same rate when compared to the \emph{Cooperative} and \emph{Non-Cooperative} schemes. A \emph{``Simultaneous-Successive Relaying based on Dirty paper coding scheme" (SSRD)} is also proposed. The optimum scheduling of the relays and hence the capacity of the half-duplex Gaussian parallel relay channel in the low and high signal-to-noise ratio (SNR) scenarios is derived. In the low SNR scenario, it is revealed that under certain conditions for the channel coefficients, the ratio of the achievable rate of the simultaneous relaying based on BCM to the cut-set bound tends to be 1. On the other hand, as SNR goes to infinity, it is proved that successive relaying, based on the DPC, asymptotically achieves the capacity of the network. Schein and Gallager introduced the Gaussian parallel relay channel in 2000. They proposed the Amplify-and-Forward (AF) and the Decode-and-Forward (DF) strategies for this channel. For a long time, the best known achievable rate for this channel was based on the AF and DF with time sharing (AF-DF). Recently, a Rematch-and-Forward (RF) scheme for the scenario in which different amounts of bandwidth can be assigned to the first and second hops were proposed. In chapter 3, we propose a \emph{Combined Amplify-and-Decode Forward (CADF)} scheme for the Gaussian parallel relay channel. We prove that the CADF scheme always gives a better achievable rate compared to the RF scheme, when there is a bandwidth mismatch between the first hop and the second hop. Furthermore, for the equal bandwidth case (Schein's setup), we show that the time sharing between the CADF and the DF schemes (CADF-DF) leads to a better achievable rate compared to the time sharing between the RF and the DF schemes (RF-DF) as well as the AF-DF.

Cooperative wireless communication has received significant attention during recent years due to several reasons. First, since the received power decreases rapidly with distance, the idea of multi-hopping is becoming of particular importance. In multi-hopped communication, the source exploits some intermediate nodes as relays. Then the source sends its message via those relays to the destination. Second, relays can emulate some kind of distributed transmit antennas to form spatial diversity and combat multi-path fading effect of the wireless channel. Parallel Relay Channel is an information theoretical model for a communication system whereby a sender aims to communicate to a receiver with the help of relay nodes. It represents the simplest model for a multi–hop wireless network and a full understanding of the limits of communication over such a channel can potentially shed light on the design of more efficient wireless networks. However, the capacity of the relay channel has been established only for few special cases and little progress has been made toward solving the general case since the early 1980s. In this dissertation, motivated by practical constraints, we study the information theoretical limits of the half-duplex Gaussian Parallel Relay channel , as well as, the transmission strategies for the parallel relay channel with bandwidth mismatch between the first and the second hops. Chapter 2 investigates the problem of communication for a network composed of two half-duplex parallel relays with additive white Gaussian noise (AWGN). There is no direct link between the source and the destination. However, the relays can communicate with each other through the channel between them. Two protocols, i.e., \emph{Simultaneous} and \emph{Successive} relaying, associated with two possible relay scheduling are proposed. The simultaneous relaying protocol is based on \emph{Broadcast-multiaccess with Common Message (BCM)} scheme. For the successive relaying protocol: (i) a \emph{Non-Cooperative} scheme based on the \emph{Dirty Paper Coding (DPC)}, and (ii) a \emph{Cooperative} scheme based on the \emph{Block Markov Encoding (BME)} are considered. The composite scheme of employing BME in \emph{at most} one relay and DPC in \emph{at least} another one is shown to achieve at least the same rate when compared to the \emph{Cooperative} and \emph{Non-Cooperative} schemes. A \emph{``Simultaneous-Successive Relaying based on Dirty paper coding scheme" (SSRD)} is also proposed. The optimum scheduling of the relays and hence the capacity of the half-duplex Gaussian parallel relay channel in the low and high signal-to-noise ratio (SNR) scenarios is derived. In the low SNR scenario, it is revealed that under certain conditions for the channel coefficients, the ratio of the achievable rate of the simultaneous relaying based on BCM to the cut-set bound tends to be 1. On the other hand, as SNR goes to infinity, it is proved that successive relaying, based on the DPC, asymptotically achieves the capacity of the network. Schein and Gallager introduced the Gaussian parallel relay channel in 2000. They proposed the Amplify-and-Forward (AF) and the Decode-and-Forward (DF) strategies for this channel. For a long time, the best known achievable rate for this channel was based on the AF and DF with time sharing (AF-DF). Recently, a Rematch-and-Forward (RF) scheme for the scenario in which different amounts of bandwidth can be assigned to the first and second hops were proposed. In chapter 3, we propose a \emph{Combined Amplify-and-Decode Forward (CADF)} scheme for the Gaussian parallel relay channel. We prove that the CADF scheme always gives a better achievable rate compared to the RF scheme, when there is a bandwidth mismatch between the first hop and the second hop. Furthermore, for the equal bandwidth case (Schein's setup), we show that the time sharing between the CADF and the DF schemes (CADF-DF) leads to a better achievable rate compared to the time sharing between the RF and the DF schemes (RF-DF) as well as the AF-DF.