Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces

Levenshtein, V.I.

doi:10.1109/18.412678

Cited by 128 publications

(104 citation statements)

References 22 publications

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“…In statistical considerations, they arose in work of Eagleson [8] and later Vere-Jones [21]. They play various roles in coding theory and combinatorics, for example, in MacWilliams' theorem on weight enumerators [17,14], and in association schemes [3,4,5].…”

Section: Introductionmentioning

confidence: 99%

Krawtchouk matrices from classical and quantum random walks

Feinsilver

Kocik

2001

Algebraic Methods in Statistics and Probability

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Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic properties of these matrices provide a very interesting and accessible example in the approach to probability theory known as quantum probability. First it is noted how the Krawtchouk matrices are connected to the classical symmetric Bernoulli random walk. And we show how to derive Krawtchouk matrices in the quantum probability context via tensor powers of the elementary Hadamard matrix. Then connections with the classical situation are shown by calculating expectation values in the quantum case.

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Section: Introductionmentioning

confidence: 99%

Krawtchouk matrices from classical and quantum random walks

Feinsilver

Kocik

2001

Algebraic Methods in Statistics and Probability

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“…Together with known bounds on the minimal dimension of linear codes of given length and dual distance [15] this gives lower bounds on f k and hence also on Rel(M, p)…”

Section: ) Is the Minimum Dimension Of A Binary Linear Code Of Lengthmentioning

confidence: 99%

On Some Polynomials Related to Weight Enumerators of Linear Codes

Barg¹

2002

SIAM J. Discrete Math.

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Abstract.A linear code can be thought of as a vector matroid represented by the columns of the code's generator matrix; a well-known result in this context is Greene's theorem on a connection of the weight polynomial of the code and the Tutte polynomial of the matroid. We examine this connection from the coding-theoretic viewpoint, building upon the rank polynomial of the code. This enables us to obtain bounds on all-terminal reliability of linear matroids and new proofs of two known results: Greene's theorem and a connection between the weight polynomial and the partition polynomial of the Potts model. Key words. all-terminal reliability, Greene's theorem, linear code, linear matroid, Potts model, rank polynomial, Tutte polynomial, weight enumerator AMS subject classifications. 94B05, 05B35PII. S0895480199364148 Introduction.A linear matroid M together with a chosen representation over a finite field F q is the same object as a linear code. The most well-known result underlining this connection is Greene's theorem on the relation of the weight polynomial of the code and the Tutte polynomial of the matroid. In this paper we further examine the relation between the polynomial invariants of codes, matroids, and some other combinatorial objects. Our point of view is coding-theoretic. We begin with listing basic definitions for linear codes and some very simple linear-algebraic properties of subcodes. These properties lead almost immediately to a relation between the weight polynomial of a linear code and the rank polynomial of the corresponding matroid. This relation is equivalent to Greene's theorem which is shown to be a purely linear-algebraic fact. An advantage of the coding-theoretic point of view is determined by the fact that the weight polynomial enjoys more structural properties than more general matroid invariants; when this structure translates to other problems, it sometimes produces interesting insights.As an example, we relate the reliability polynomial of a linear matroid to an evaluation of the weight polynomial of the code. The corresponding functional on linear codes turns out to be well studied under the name of the probability of undetected error of the code. Together with some related ideas this enables us to derive upper and lower bounds on the matroid reliability. As another application of the weightrank connection, we give a direct proof of the link between the partition function of the Potts model and the weight polynomial of the cocycle code of the graph.General sources for coding theory are the books [17], [19]. Relevant applications of the Tutte polynomial are covered in [6], [24]. All the necessary information on interaction models is contained in [24].

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“…The case p = q = 1 2 is trivial and we will not consider it here. It is well-known that the properties of zeros of Krawtchouk polynomials are important in the study of the Hamming scheme of classical coding theory; see [4,14,16,21]. Also, Lloyd's theorem [13,16] states that the existence of a perfect code in the Hamming metric corresponds to the Krawtchouk polynomials having integer zeros.…”

Section: Introductionmentioning

confidence: 99%

Global Asymptotics of Krawtchouk Polynomials – a Riemann-Hilbert Approach*

Dai

Wong

2007

Chin. Ann. Math. Ser. B

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In this paper, we study the asymptotics of the Krawtchouk polynomials K N n (z; p, q) as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters n/N tends to a limit c ∈ (0, 1) as n → ∞. The results are globally valid in one or two regions in the complex z-plane depending on the values of c and p; in particular, they are valid in regions containing the interval on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou.2000 Mathematics Subject Classification. Primary 41A60, 33C45.

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Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces

Cited by 128 publications

References 22 publications

Krawtchouk matrices from classical and quantum random walks

Krawtchouk matrices from classical and quantum random walks

On Some Polynomials Related to Weight Enumerators of Linear Codes

Global Asymptotics of Krawtchouk Polynomials – a Riemann-Hilbert Approach*

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