Applications of ordered weights in information transmission

This dissertation is devoted to a study of a class of linear codes related to a particular metric space that generalizes the Hamming space in that the metric function is defined by a partial order on the set of coordinates of the vector. We begin with developing combinatorial and linear-algebraic aspects of linear ordered codes. In particular, we define multivariate rank enumerators for linear codes and show that they form a natural set of invariants in the study of the duality of linear codes. The rank enumerators are further shown to be connected to the shape distributions of linear codes, and enable us to give a simple proof of a MacWilliams-like theorem for the ordered case. We also pursue the connection between linear codes and matroids in the ordered case and show that the rank enumerator can be thought of as an instance of the classical matroid invariant called the Tutte polynomial. Finally, we consider the distributions of support weights of ordered codes and their expression via the rank enumerator. Altogether, these results generalize a group of well-known results for codes in the Hamming space to the ordered case. Extending the research in the first part, we define simple probabilistic channel models that are in a certain sense matched to the ordered distance, and prove several results related to performance of linear codes on such channels. In particular, we define ordered wire-tap channels and establish several results related to the use of linear codes for reliable and secure transmission in such channel models. In the third part of this dissertation we study polar coding schemes for channels with nonbinary input alphabets. We construct a family of linear codes that achieve the capacity of a nonbinary symmetric discrete memoryless channel with input alphabet of size q=2^r, r=2,3,.... A new feature of the coding scheme that arises in the nonbinary case is related to the emergence of several extremal configurations for the polarized data symbols. We establish monotonicity properties of the configurations and use them to show that total transmission rate approaches the symmetric capacity of the channel. We develop these results to include the case of ``controlled polarization'' under which the data symbols polarize to any predefined set of extremal configurations. We also outline an application of this construction to data encoding in video sequences of the MPEG-2 and H.264/MPEG-4 standards

Bounds on the size of codes

In this dissertation we determine new bounds and properties of codes in three different finite metric spaces, namely the ordered Hamming space, the binary Hamming space, and the Johnson space. The ordered Hamming space is a generalization of the Hamming space that arises in several different problems of coding theory and numerical integration. Structural properties of this space are well described in the framework of Delsarte's theory of association schemes. Relying on this theory, we perform a detailed study of polynomials related to the ordered Hamming space and derive new asymptotic bounds on the size of codes in this space which improve upon the estimates known earlier. A related project concerns linear codes in the ordered Hamming space. We define and analyze a class of near-optimal codes, called near-Maximum Distance Separable codes. We determine the weight distribution and provide constructions of such codes. Codes in the ordered Hamming space are dual to a certain type of point distributions in the unit cube used in numerical integration. We show that near-Maximum Distance Separable codes are equivalently represented as certain near-optimal point distributions. In the third part of our study we derive a new upper bound on the size of a family of subsets of a finite set with restricted pairwise intersections, which improves upon the well-known Frankl-Wilson upper bound. The new bound is obtained by analyzing a refinement of the association scheme of the Hamming space (the Terwilliger algebra) and intertwining functions of the symmetric group. Finally, in the fourth set of problems we determine new estimates on the size of codes in the Johnson space. We also suggest a new approach to the derivation of the well-known Johnson bound for codes in this space. Our estimates are often valid in the region where the Johnson bound is vacuous. We show that these methods are also applicable to the case of multiple packings in the Hamming space (list-decodable codes). In this context we recover the best known estimate on the size of list-decodable codes in a new way

Translation association schemes, poset metrics, and the shape enumerator of codes

Poset metrics form a generalization of the Hamming metric on the space double-struck F q n. Orbits of the group of linear isometries of the space give rise to a translation association scheme. The structure of the dual scheme is important in studying duality of linear codes; this study is facilitated if the scheme is self-dual. We study the relation between self-duality of the scheme and that of the poset. We also give new examples of poset metric spaces and describe the association schemes that arise from linear isometries. © 2012 IEEE.Poset metrics form a generalization of the Hamming metric on the space double-struck F q n. Orbits of the group of linear isometries of the space give rise to a translation association scheme. The structure of the dual scheme is important in studying dual101105sem informaçãosem informaçãoBarg, A., Purkayastha, P., Bounds on ordered codes and orthogonal arrays (2009) Moscow Mathematical Journal, 9 (2), pp. 211-243. , arXiv:cs/0702033Brouwer, A.E., Cohen, A.M., Neumaier, A., (1989) Distance-regular Graphs, , Springer-Verlag, Berlin e. aBrualdi, R.A., Graves, J.S., Lawrence, K.M., Codes with a poset metric (1995) Discrete Math., 147 (1-3), pp. 57-72Dougherty, S.T., Skriganov, M.M., MacWilliams duality and the Rosenbloom-Tsfasman metric (2002) Mosc. Math. J., 2 (1), pp. 81-97+199Kim, D.S., Kim, H.K., Duality of Translation Association Schemes Coming from Certain Actions, , arXiv:1108.4947Kim, H.K., Oh, D.Y., A classification of posets admitting the MacWilliams identity (2005) IEEE Trans. Inform. Theory, 51 (4), pp. 1424-1431Lee, K., The automorphism group of a linear space with the Rosenbloom-Tsfasman metric (2003) Eur. J. Combinatorics, 24, pp. 607-612Martin, W.J., Stinson, D.R., Association schemes for ordered orthogonal arrays and (T,M, S)-nets (1999) Canad. J. Math., 51 (2), pp. 326-346Panek, L., Firer, M., Kim, H.K., Hyun, J.Y., Groups of linear isometries on poset structures (2008) Discrete Math., 308 (18), pp. 4116-4123Park, W., Barg, A., Linear ordered codes, shape enumerators, and parallel channels Proc. 48th Annual Allerton Conf. Commun. Control Comput., Monticello, IL, Sept. 30-Oct. 1, 2010, pp. 361-367Rosenbloom, M.Yu., Tsfasman, M.A., Codes for the m-metric (1997) Problems of Information Transmission, 33 (1), pp. 45-52IEEE International Symposium on Information Theory ProceedingsResearch of this author supported in part by NSF grants CCF0916919 and DMS110169

A comparison of the CAR and DAGAR spatial random effects models with an application to diabetics rate estimation in Belgium

When hierarchically modelling an epidemiological phenomenon on a finite collection of sites in space, one must always take a latent spatial effect into account in order to capture the correlation structure that links the phenomenon to the territory. In this work, we compare two autoregressive spatial models that can be used for this purpose: the classical CAR model and the more recent DAGAR model. Differently from the former, the latter has a desirable property: its ρ parameter can be naturally interpreted as the average neighbor pair correlation and, in addition, this parameter can be directly estimated when the effect is modelled using a DAGAR rather than a CAR structure. As an application, we model the diabetics rate in Belgium in 2014 and show the adequacy of these models in predicting the response variable when no covariates are available