The definition of polar codes given by Arikan is explicit, but the construction complexity is an issue. This is due to the exponential growth in the size of the output alphabet of the bit-channels as the codeword length increases. Tal and Vardy recently presented a method for constructing polar codes which controls this growth. They approximated each bit-channel with a “better” channel and a “worse” channel while reducing the alphabet size. They constructed a polar code based on the “worse” channel and used the “better” channel to measure the distance from the optimal channel. This thesis considers the knowledge gained from the perspective of the “better” channel. A method is presented using iterative upgrading of the bit-channels which successively results in a channel closer to the original one. It is shown that this approach can be used to obtain a channel arbitrarily close to the original channel, and therefore to the optimal construction of a polar code. / Graduate / 0984 / 0544 / arash.ghayoori@gmail.com

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.

We study the Hamming distance and the structure of repeated-root cyclic codes, and their generalizations to constacyclic and polycyclic codes, over finite fields and Galois rings. We develop a method to compute the Hamming distance of these codes. Our computation gives the Hamming distance of constacyclic codes of length $np^s$ in many cases. In particular, we determine the Hamming distance of all constacyclic, and therefore cyclic and negacyclic, codes of lengths p^s and 2p^s over a finite field of characteristic $p$. It turns out that the generating sets for the ambient space obtained by torsional degrees and strong Groebner basis for the ambient space are essentially the same and one can be obtained from the other. In the second part of the thesis, we study matrix product codes. We show that using nested constituent codes and a non-constant matrix in the construction of matrix product codes with polynomial units is a crucial part of the construction. We prove a lower bound on the Hamming distance of matrix product codes with polynomial units when the constituent codes are nested. This generalizes the technique used to construct the record-breaking examples of Hernando and Ruano. Contrary to a similar construction previously introduced, this bound is not sharp and need not hold when the constituent codes are not nested. We give a comparison of this construction with a previous one. We also construct new binary codes having the same parameters, of the examples of Hernando and Ruano, but non-equivalent to them.

This thesis presents the development of a Sensor Tower Platform (STP) comprised of an array of Passive Infra-Red (PIR) sensors along with a classification algorithm that enables the STP to distinguish between human intrusion, animal intrusion and clutter arising from wind-blown vegetative movement in an outdoor environment. The research was motivated by the aim of exploring the potential use of wireless sensor networks (WSNs) as an early-warning system to help mitigate human-wildlife conflicts occurring at the edge of a forest. While PIR sensors are in commonplace use in indoor settings, their use in an outdoor environment is hampered by the fact that they are prone to false alarms arising from wind-blown vegetation. Every PIR sensor is made up of one or more pairs of pyroelectric pixels arranged in a plane, and the orientation of interest in this thesis is one in which this plane is a vertical plane, i.e., a plane perpendicular to the ground plane. The intersection of the Field Of View (FOV) of the PIR sensor with a second vertical plane that lies within the FOV of the PIR sensor, is called the virtual pixel array (VPA). The structure of the VPA corresponding to the plane along which intruder motion takes place determines the form of the signal generated by the PIR sensor. The STP developed in this thesis employs an array of PIR sensors designed so as to result in a VPA that makes it easier to discriminate between human and animal intrusion while keeping to a small level false alarms arising from vegetative motion. The design was carried out in iterative fashion, with each successive iteration separated by a lengthy testing phase. There were a total of 5 design iterations spanning a total period of 14 months. Given the inherent challenges involved in gathering data corresponding to animal intrusion, the testing of the SP was carried out both using real-world data and through simulation. Simulation was carried out by developing a tool that employed animation software to simulate intruder and animal motion as well as some limited models of wind-blown vegetation. More specifically, the simulation tool employed 3-dimensional models of intruder and shrub motion that were developed using the popular animation software Blender. The simulated output signal of the PIR sensor was then generated by calculating the area of the 3-dimensional intruder when projected onto the VPA of the STP. An algorithm for efficiently calculating this to a good degree of approximation was implemented in Open Graphics Library (OpenGL). The simulation tool was useful both for evaluating various competing design alternatives as well as for developing an intuition for the kind of signals the SP would generate without the need for time-consuming and challenging animal-motion data collection. Real-world data corresponding to human motion was gathered on the campus of the Indian Institute of Science (IISc), while animal data was recorded at a dog-trainer facility in Kengeri as well as the Bannerghatta Biological Park, both located in the outskirts of Bengaluru. The array of PIR sensors was designed so as to result in a VPA that had good spatial resolution. The spatial resolution capabilities of the STP permitted distinguishing between human and animal motion with good accuracy based on low-complexity, signal-energy computations. Rejecting false alarms arising from vegetative movement proved to be more challenging. While the inherent spatial resolution of the STP was very helpful, an alternative approach turned out to have much higher accuracy, although it is computationally more intensive. Under this approach, the intruder signal, either human or animal, was modelled as a chirp waveform. When the intruder moves along a circular arc surrounding the STP, the resulting signal is periodic with constant frequency. However, when the intruder moves along a more likely straight-line path, the resultant signal has a strong chirp component. Clutter signals arising from vegetative motion does not exhibit this chirp behavior and an algorithm that exploited this difference turned in a classification accuracy in excess of 97%.