RESEARCH STATEMENT PUNARBASU PURKAYASTHA

+1-301-405-6621 (Office) 

[email protected]



http://www.ece.umd.edu/∼ppurka

My areas of expertise include coding theory, combinatorics (algebraic and enumerative aspects), theory of orthogonal polynomials, and optimization. The subject of coding theory has traditionally focused on many wide and varied aspects of information transmission, storage and retrieval. In the last decade it has expanded into applications to computer science (testability of Boolean functions, sampling, numerical integration, etc.), computational biology, cryptography, and discrete geometry. Coding theory has been a vibrant and evolving research area both as a division of applied mathematics and as an engineering discipline. In my research I have primarily focused on extremal problems in coding theory (bounds on codes), using ideas from various areas of applied mathematics to derive new properties and estimates of the size of codes in finite metric spaces. I have studied bounds on list decoding of codes and have provided new proofs and interpretation of the well-known Johnson bound in this context. I have also extended the applicability range of the Johnson bound by proving new upper estimates on the size of constant weight codes in the binary Hamming space. Additionally, I have studied properties of packings of the “ordered Hamming space.” Such packings, called ordered codes, are closely related to distributions of points in the unit cube, used for numerical integration of functions. In particular, I have • established new properties of the association scheme supported by the ordered Hamming space; • established new upper bounds on the size of ordered codes; • studied near-Maximum Distance Separable codes in the ordered Hamming space, determined their weight distribution, and their relation to distributions of points in the unit cube. Accomplished Research New upper bounds on constant weight codes in the binary Hamming space: The Johnson bound for constant weight codes in the binary Hamming space is a well known upper bound. It is typically proved by averaging over all the distances in the constant weight code. In [5] I showed that this averaging technique can be adapted to provide sharper bounds on constant weight codes. In particular, I provided two new upper bounds which improve the existing Johnson bound for a varied range of parameters. The first bound is obtained by considering a weighted average of the distances in the constant weight code. The second bound is obtained by observing that the L2 norm (with uniform measure) of a sequence of numbers is at least as large as the L1 norm (with uniform measure) of the sequence. These new bounds are also valid in the region at or beyond the Johnson radius where the Johnson bound does not exist. The values of these new bounds are sometimes exact and meet the table of bounds for constant weight codes presented in Agrell, Vardy and Zeger [1]. The techniques I have introduced can potentially be adapted to improve non-asymptotic upper bounds on constant weight codes in the q-ary Hamming space (in this case, such a generalization is far from straightforward). Bounds on codes in the ordered Hamming space: The ordered Hamming space, also known as the Niederreiter-Rosenbloom-Tsfasman space, was introduced by Niederreiter [8] and independently by Rosenbloom and Tsfasman [9]. Niederreiter introduced this metric in the study of distributions of point sets in the unit cube called (t, m, s)-nets. (t, m, s)-nets are of interest because of their application to quasi-Monte-Carlo integration of functions defined on the unit cube. Rosenbloom and Tsfasman introduced this metric in order to describe a new communication channel. The work of Martin and Stinson [7] established that certain objects called ordered orthogonal arrays (OOAs), which are closely related to (t, m, s)-nets, are dual objects to ordered 1

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codes, in the context of Delsarte’s theory of Association Schemes [6]. Bounds on OOAs and (t, m, s)-nets are of interest for estimating the error of quasi-Monte-Carlo integration. In [3] I derived new finite-length and asymptotic upper bounds on the size of ordered codes of a given minimum distance, namely, an Elias bound and asymptotic Linear Programming (LP) bounds. By the duality between OOAs and ordered codes I have also obtained lower bounds on OOAs and (t, m, s)-nets. As part of the derivation of the Elias bound, I determined the Johnson bound on constant weight codes in the ordered Hamming space. The LP bound was obtained by application of the theory of Association Schemes. In particular, I studied properties of multivariate Krawtchouk polynomials which are associated with the ordered Hamming scheme, established a 3-term recurrence relation for these polynomials, and determined the Christoffel-Darboux formula. Traditionally, the LP bound for codes in Hamming spaces was obtained by determining the extremal roots of univariate Krawtchouk polynomials. Owing to the difficulty in estimating the extremal roots of multivariate Krawtchouk polynomials, I instead applied the “Spectral Method”, first introduced by Bachoc [2] in the context of LP bounds for Grassmannian spaces, thereby obtaining a universal bound on the size of ordered codes and OOAs. For the parameters of the ordered Hamming space when the arising Krawtchouk polynomials are bivariate, I was able to derive the behavior of the extremal roots, which resulted in an improved LP bound. The results obtained give the best known asymptotic upper bounds on the size of ordered codes. Linear codes in the ordered Hamming Space: Codes which have parameters close to or at the theoretical upper bounds are of much interest since they are optimal (or close to optimal) for their values of the minimum distance. Maximum Distance Separable (MDS) codes are a famous example of such codes, whose minimum distance attains the Singleton upper bound. MDS codes in the Hamming space are connected with some classical problems in finite geometries related to the existence of some extremal configurations in projective geometries. MDS codes in the ordered Hamming space are equivalent to “optimal distributions” of points in the unit cube [10]. In this context, I have considered near-Maximum Distance Separable (NMDS) codes whose minimum distance is one unit away from the Singleton upper bound. In [4] I have shown that the concept of linear NMDS codes and generalized Hamming weights can be readily generalized to the ordered Hamming space, and that many of their properties can be proved via linear algebraic arguments. I have determined the weight distribution of NMDS codes and proved that NMDS codes are equivalently represented as certain distributions of points in the unit cube. To prove this set of results I studied the properties of the NMDS code simultaneously as a linear ordered code and as a linear ordered orthogonal array, which involves interesting combinatorial considerations. References [1] E. Agrell, A. Vardy, and K. Zeger, Upper bounds for constant-weight codes, IEEE Trans. Inform. Theory 46 (2000), 2373–2395. [2] C. Bachoc, Linear programming bounds for codes in Grassmannian spaces, IEEE Trans. Inform. Theory 52 (2006), 2111–2126. [3] A. Barg and P. Purkayastha, Bounds on ordered codes and orthogonal arrays, Moscow Mathematical Journal 9 (2009), no. 2, 211–243. [4] A. Barg and P. Purkayastha, Near MDS poset codes and distributions, preprint, submitted to Error-Correcting Codes, Cryptography and Finite Geometries, Editors: A. Bruen and D. Wehlau, AMS series in Contemporary Mathematics. [5] A. Barg and P. Purkayastha, New upper bounds on constant weight codes, preprint, to be submitted to IEEE Trans. Inform. Theory. [6] P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Repts. Suppl. 10 (1973), 1–97. [7] W. J. Martin and D. R. Stinson, Association schemes for ordered orthogonal arrays and (T, M, S)-nets, Canad. J. Math. 51 (1999), no. 2, 326–346. [8] H. Niederreiter, Low-discrepancy point sets, Monatsh. Math. 102 (1986), no. 2, 155–167. [9] M. Yu. Rosenbloom and M. A. Tsfasman, Codes for the m-metric, Problems of Information Transmission 33 (1997), no. 1, 45–52. [10] M. M. Skriganov, Coding theory and uniform distributions, Algebra i Analiz 13 (2001), no. 2, 191–239, English translation in St. Petersburg Math. J. vol. 13 (2002), no. 2, 301–337.