Survey of binary Krawtchouk
                    polynomials

Krasikov, Ilia; Litsyn, Simon

doi:10.1090/dimacs/056/16

Cited by 32 publications

(34 citation statements)

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“…We have seen that the f i (x) satisfy the recurrence relation (5). On the other hand, it is known (see, for example, [4]) that…”

Section: Remark 41 the Binary Krawtchouk Polynomial P N I (X) Is Dementioning

confidence: 92%

Switching reconstruction of digraphs

Bondy¹,

Mercier²

2011

J. Graph Theory

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Abstract:We introduce the problem of reconstructing a directed graph from the family of digraphs obtained by reversing the arcs at each vertex. Our approach is elementary, and applies to a variety of switching reconstruction problems, including the classical question for undirected graphs introduced by R. P. Stanley in 1985. It provides inter alia an alternative proof of the theorem of M. N. Ellingham and G. Royle on the switching reconstruction of subgraphs. One significant difference between the directed and undirected problems is that there exist switching-nonreconstructible directed graphs on eight vertices. ᭧

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“…We have seen that the f i (x) satisfy the recurrence relation (5). On the other hand, it is known (see, for example, [4]) that…”

Section: Remark 41 the Binary Krawtchouk Polynomial P N I (X) Is Dementioning

confidence: 92%

Switching reconstruction of digraphs

Bondy¹,

Mercier²

2011

J. Graph Theory

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“…We make use of Krawtchouk polynomials for probability p. These polynomials are orthogonal with respect to the binomial distribution (see [39] for the general definition, and [29] for the special case p = 1/2, which we use in Sections 5 and 6). We treat them as column vectors in R m+1 and also include the weight (due to the weight, they cease to be polynomials).…”

Section: Application Of Representation Theorymentioning

confidence: 99%

Quantum Algorithms for Learning Symmetric Juntas via Adversary Bound

Belovs

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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In this paper, we study the following variant of the junta learning problem. We are given oracle access to a Boolean function f on n variables that only depends on k variables, and, when restricted to them, equals some predefined function h. The task is to identify the variables the function depends on. When h is the XOR or the OR function, this gives a restricted variant of the Bernstein-Vazirani or the combinatorial group testing problem, respectively.We analyse the general case using the adversary bound, and give an alternative formulation for the quantum query complexity of this problem. We construct optimal quantum query algorithms for the cases when h is the OR function (complexity is Θ( √ k)) or the exact-half function (complexity is Θ(k 1/4 )). The first algorithm resolves an open problem from [4]. For the case when h is the majority function, we prove an upper bound of O(k 1/4 ). All these algorithms can be made exact. We obtain a quartic improvement when compared to the randomised complexity (if h is the exacthalf or the majority function), and a quadratic one when compared to the non-adaptive quantum complexity (for all functions considered in the paper).

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“…Also, the null space of matrix is now restricted to linear functions instead of affine functions. Then, equality (6) shows that for odd , since is an affine function. For even still keeps the single (nonzero) function .…”

Section: A Encoding and Code Dimensionmentioning

confidence: 99%

“…This obviously gives the matrix (11) Next, we prove that the last row in the matrix is linearly dependent of the first rows. Indeed, observe that either of two vectors defined in (6) gives the all-zero code vector on one of the two submatrices (11) and the all-one vector on the other. Namely, for odd , we obtain the last row in by taking…”

Section: B Recursive Representationmentioning

confidence: 99%

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Spherically Punctured Biorthogonal Codes

Dumer

Kapralova

2013

IEEE Trans. Inform. Theory

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Consider a binary Reed-Muller code defined on the hypercube and let all code positions be restricted to the -tuples of a given Hamming weight . In this paper, we specify this single-layer construction obtained from the biorthogonal codes and the Hadamard codes . Both punctured codes inherit some recursive properties of the original RM codes; however, they cannot be formed by the recursive Plotkin construction. We first observe that any code vector in these codes has Hamming weight defined by the weight of its information block. More specifically, this weight depends on the absolute values of the Krawtchouk polynomials . We then study the properties of the Krawtchouk polynomials and show that the minimum code weight of a single-layer code is achieved at the minimum input weight for any . We further refine our codes by limiting the possible weights of the input information blocks. As a result, some of the designed code sequences meet or closely approach the Griesmer bound. Finally, we consider more general punctured codes whose positions form several spherical layers.

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Survey of binary Krawtchouk polynomials

Cited by 32 publications

References 0 publications

Switching reconstruction of digraphs

Switching reconstruction of digraphs

Quantum Algorithms for Learning Symmetric Juntas via Adversary Bound

Spherically Punctured Biorthogonal Codes

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