Divisible Codes - A Survey

This paper surveys parts of the study of divisibility properties of codes. The survey begins with the motivating background involving polynomials over finite fields. Then it presents recent results on bounds and applications to optimal codes.

Error correcting codes for visible light communication systems

Over the past few years, the number of wireless networks users has been increasing. Until now, Radio-Frequency (RF) used to be the dominant technology. However, the electromagnetic spectrum in these region is being saturated, demanding for alternative wireless technologies. Recently, with the growing market of LED lighting, the Visible Light Communications has been drawing attentions from the research community. First, it is an eficient device for illumination. Second, because of its easy modulation and high bandwidth. Finally, it can combine illumination and communication in the same device, in other words, it allows to implement highly eficient wireless communication systems. One of the most important aspects in a communication system is its reliability when working in noisy channels. In these scenarios, the received data can be afected by errors. In order to proper system working, it is usually employed a Channel Encoder in the system. Its function is to code the data to be transmitted in order to increase system performance. It commonly uses ECC, which appends redundant information to the original data. At the receiver side, the redundant information is used to recover the erroneous data. This dissertation presents the implementation steps of a Channel Encoder for VLC. It was consider several techniques such as Reed-Solomon and Convolutional codes, Block and Convolutional Interleaving, CRC and Puncturing. A detailed analysis of each technique characteristics was made in order to choose the most appropriate ones. Simulink models were created in order to simulate how diferent codes behave in diferent scenarios. Later, the models were implemented in a FPGA and simulations were performed. Hardware co-simulations were also implemented to faster simulation results. At the end, diferent techniques were combined to create a complete Channel Encoder capable of detect and correct random and burst errors, due to the usage of a RS(255,213) code with a Block Interleaver. Furthermore, after the decoding process, the proposed system can identify uncorrectable errors in the decoded data due to the CRC-32 algorithm.

Evaluating the specificity of community injury hospitalization data over time

This study identified the areas of poor specificity in national injury hospitalization data and the areas of improvement and deterioration in specificity over time. A descriptive analysis of ten years of national hospital discharge data for Australia from July 2002-June 2012 was performed. Proportions and percentage change of defined/undefined codes over time was examined. At the intent block level, accidents and assault were the most poorly defined with over 11% undefined in each block. The mechanism blocks for accidents showed a significant deterioration in specificity over time with up to 20% more undefined codes in some mechanisms. Place and activity were poorly defined at the broad block level (43% and 72% undefined respectively). Private hospitals and hospitals in very remote locations recorded the highest proportion of undefined codes. Those aged over 60 years and females had the higher proportion of undefined code usage. This study has identified significant, and worsening, deficiencies in the specificity of coded injury data in several areas. Focal attention is needed to improve the quality of injury data, especially on those identified in this study, to provide the evidence base needed to address the significant burden of injury in the Australian community.

A unified construction of space-time codes with optimal rate-diversity tradeoff

The problem of constructing space-time (ST) block codes over a fixed, desired signal constellation is considered. In this situation, there is a tradeoff between the transmission rate as measured in constellation symbols per channel use and the transmit diversity gain achieved by the code. The transmit diversity is a measure of the rate of polynomial decay of pairwise error probability of the code with increase in the signal-to-noise ratio (SNR). In the setting of a quasi-static channel model, let n(t) denote the number of transmit antennas and T the block interval. For any n(t) <= T, a unified construction of (n(t) x T) ST codes is provided here, for a class of signal constellations that includes the familiar pulse-amplitude (PAM), quadrature-amplitude (QAM), and 2(K)-ary phase-shift-keying (PSK) modulations as special cases. The construction is optimal as measured by the rate-diversity tradeoff and can achieve any given integer point on the rate-diversity tradeoff curve. An estimate of the coding gain realized is given. Other results presented here include i) an extension of the optimal unified construction to the multiple fading block case, ii) a version of the optimal unified construction in which the underlying binary block codes are replaced by trellis codes, iii) the providing of a linear dispersion form for the underlying binary block codes, iv) a Gray-mapped version of the unified construction, and v) a generalization of construction of the S-ary case corresponding to constellations of size S-K. Items ii) and iii) are aimed at simplifying the decoding of this class of ST codes.

Queued Cooperative Wireless Networks With Rateless Codes

Cooperative communication using rateless codes, in which the source transmits an infinite number of parity bits to the destination until the receipt of an acknowledgment, has recently attracted considerable interest. It provides a natural and efficient mechanism for accumulating mutual information from multiple transmitting relays. We develop an analysis of queued cooperative relay systems that combines the communication-theoretic transmission aspects of cooperative communication using rateless codes over Rayleigh fading channels with the queuing-theoretic aspects associated with buffering messages at the relays. Relay cooperation combined with queuing reduces the message transmission times and also helps distribute the traffic load in the network, which improves throughput significantly.

On erasure coding for distributed storage and streaming communications

The work presented in this thesis revolves around erasure correction coding, as applied to distributed data storage and real-time streaming communications.

First, we examine the problem of allocating a given storage budget over a set of nodes for maximum reliability. The objective is to find an allocation of the budget that maximizes the probability of successful recovery by a data collector accessing a random subset of the nodes. This optimization problem is challenging in general because of its combinatorial nature, despite its simple formulation. We study several variations of the problem, assuming different allocation models and access models, and determine the optimal allocation and the optimal symmetric allocation (in which all nonempty nodes store the same amount of data) for a variety of cases. Although the optimal allocation can have nonintuitive structure and can be difficult to find in general, our results suggest that, as a simple heuristic, reliable storage can be achieved by spreading the budget maximally over all nodes when the budget is large, and spreading it minimally over a few nodes when it is small. Coding would therefore be beneficial in the former case, while uncoded replication would suffice in the latter case.

Second, we study how distributed storage allocations affect the recovery delay in a mobile setting. Specifically, two recovery delay optimization problems are considered for a network of mobile storage nodes: the maximization of the probability of successful recovery by a given deadline, and the minimization of the expected recovery delay. We show that the first problem is closely related to the earlier allocation problem, and solve the second problem completely for the case of symmetric allocations. It turns out that the optimal allocations for the two problems can be quite different. In a simulation study, we evaluated the performance of a simple data dissemination and storage protocol for mobile delay-tolerant networks, and observed that the choice of allocation can have a significant impact on the recovery delay under a variety of scenarios.

Third, we consider a real-time streaming system where messages created at regular time intervals at a source are encoded for transmission to a receiver over a packet erasure link; the receiver must subsequently decode each message within a given delay from its creation time. For erasure models containing a limited number of erasures per coding window, per sliding window, and containing erasure bursts whose maximum length is sufficiently short or long, we show that a time-invariant intrasession code asymptotically achieves the maximum message size among all codes that allow decoding under all admissible erasure patterns. For the bursty erasure model, we also show that diagonally interleaved codes derived from specific systematic block codes are asymptotically optimal over all codes in certain cases. We also study an i.i.d. erasure model in which each transmitted packet is erased independently with the same probability; the objective is to maximize the decoding probability for a given message size. We derive an upper bound on the decoding probability for any time-invariant code, and show that the gap between this bound and the performance of a family of time-invariant intrasession codes is small when the message size and packet erasure probability are small. In a simulation study, these codes performed well against a family of random time-invariant convolutional codes under a number of scenarios.

Finally, we consider the joint problems of routing and caching for named data networking. We propose a backpressure-based policy that employs virtual interest packets to make routing and caching decisions. In a packet-level simulation, the proposed policy outperformed a basic protocol that combines shortest-path routing with least-recently-used (LRU) cache replacement.

Gröbner basis techniques for certain problems in coding and systems theory

There is much common ground between the areas of coding theory and systems theory. Fitzpatrick has shown that a Göbner basis approach leads to efficient algorithms in the decoding of Reed-Solomon codes and in scalar interpolation and partial realization. This thesis simultaneously generalizes and simplifies that approach and presents applications to discrete-time modeling, multivariable interpolation and list decoding. Gröbner basis theory has come into its own in the context of software and algorithm development. By generalizing the concept of polynomial degree, term orders are provided for multivariable polynomial rings and free modules over polynomial rings. The orders are not, in general, unique and this adds, in no small way, to the power and flexibility of the technique. As well as being generating sets for ideals or modules, Gröbner bases always contain a element which is minimal with respect tot the corresponding term order. Central to this thesis is a general algorithm, valid for any term order, that produces a Gröbner basis for the solution module (or ideal) of elements satisfying a sequence of generalized congruences. These congruences, based on shifts and homomorphisms, are applicable to a wide variety of problems, including key equations and interpolations. At the core of the algorithm is an incremental step. Iterating this step lends a recursive/iterative character to the algorithm. As a consequence, not all of the input to the algorithm need be available from the start and different "paths" can be taken to reach the final solution. The existence of a suitable chain of modules satisfying the criteria of the incremental step is a prerequisite for applying the algorithm.

Simplified decoding techniques for linear block codes

Error correcting codes are combinatorial objects, designed to enable reliable transmission of digital data over noisy channels. They are ubiquitously used in communication, data storage etc. Error correction allows reconstruction of the original data from received word. The classical decoding algorithms are constrained to output just one codeword. However, in the late 50’s researchers proposed a relaxed error correction model for potentially large error rates known as list decoding. The research presented in this thesis focuses on reducing the computational effort and enhancing the efficiency of decoding algorithms for several codes from algorithmic as well as architectural standpoint. The codes in consideration are linear block codes closely related to Reed Solomon (RS) codes. A high speed low complexity algorithm and architecture are presented for encoding and decoding RS codes based on evaluation. The implementation results show that the hardware resources and the total execution time are significantly reduced as compared to the classical decoder. The evaluation based encoding and decoding schemes are modified and extended for shortened RS codes and software implementation shows substantial reduction in memory footprint at the expense of latency. Hermitian codes can be seen as concatenated RS codes and are much longer than RS codes over the same aphabet. A fast, novel and efficient VLSI architecture for Hermitian codes is proposed based on interpolation decoding. The proposed architecture is proven to have better than Kötter’s decoder for high rate codes. The thesis work also explores a method of constructing optimal codes by computing the subfield subcodes of Generalized Toric (GT) codes that is a natural extension of RS codes over several dimensions. The polynomial generators or evaluation polynomials for subfield-subcodes of GT codes are identified based on which dimension and bound for the minimum distance are computed. The algebraic structure for the polynomials evaluating to subfield is used to simplify the list decoding algorithm for BCH codes. Finally, an efficient and novel approach is proposed for exploiting powerful codes having complex decoding but simple encoding scheme (comparable to RS codes) for multihop wireless sensor network (WSN) applications.

Generator Matrix Based Search for Extremal Self-Dual Binary Error-Correcting Codes

Self-dual doubly even linear binary error-correcting codes, often referred to as Type II codes, are codes closely related to many combinatorial structures such as 5-designs. Extremal codes are codes that have the largest possible minimum distance for a given length and dimension. The existence of an extremal (72,36,16) Type II code is still open. Previous results show that the automorphism group of a putative code C with the aforementioned properties has order 5 or dividing 24. In this work, we present a method and the results of an exhaustive search showing that such a code C cannot admit an automorphism group Z6. In addition, we present so far unpublished construction of the extended Golay code by P. Becker. We generalize the notion and provide example of another Type II code that can be obtained in this fashion. Consequently, we relate Becker's construction to the construction of binary Type II codes from codes over GF(2^r) via the Gray map.

A parallel convolutional coder including embedded puncturing with application to consumer devices

As consumers demand more functionality) from their electronic devices and manufacturers supply the demand then electrical power and clock requirements tend to increase, however reassessing system architecture can fortunately lead to suitable counter reductions. To maintain low clock rates and therefore reduce electrical power, this paper presents a parallel convolutional coder for the transmit side in many wireless consumer devices. The coder accepts a parallel data input and directly computes punctured convolutional codes without the need for a separate puncturing operation while the coded bits are available at the output of the coder in a parallel fashion. Also as the computation is in parallel then the coder can be clocked at 7 times slower than the conventional shift-register based convolutional coder (using DVB 7/8 rate). The presented coder is directly relevant to the design of modern low-power consumer devices

Locally nilpotent groups of units in tiled rings

We consider locally nilpotent subgroups of units in basic tiled rings A, over local rings O which satisfy a weak commutativity condition. Tiled rings are generalizations of both tiled orders and incidence rings. If, in addition, O is Artinian then we give a complete description of the maximal locally nilpotent subgroups of the unit group of A up to conjugacy. All of them are both nilpotent and maximal Engel. This generalizes our description of such subgroups of upper-triangular matrices over O given in M. Dokuchaev, V. Kirichenko, and C. Polcino Milies (2005) [3]. (C) 2010 Elsevier Inc. All rights reserved.

Encoding through generalized polynomial codes

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A formalized optimum code search for q-ary trellis codes

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Concording EU trade and production data over time. National Bank of Belgium Working Paper No. 239, December 2012

This paper provides concordance procedures for product-level trade and production data in the EU and examines the implications of changing product classifications on measured product adding and dropping at Belgian firms. Using the algorithms developed by Pierce and Schott (2012a, 2012b), the paper develops concordance procedures that allow researchers to trace changes in coding systems over time and to translate product-level production and trade data into a common classification that is consistent both within a single year and over time. Separate procedures are created for the eightdigit Combined Nomenclature system used to classify international trade activities at the product level within the European Union as well as for the eight-digit Prodcom categories used to classify products in European domestic production data. The paper further highlights important differences in coverage between the Prodcom and Combined Nomenclature classifications which need to be taken into account when generating combined domestic production and international trade data at the product level. The use of consistent product codes over time results in less product adding and dropping at continuing firms in the Belgian export and production data.