On Integral Zeros of Krawtchouk Polynomials

So far there is no systematic attempt to construct Boolean functions with maximum annihilator immunity. In this paper we present a construction keeping in mind the basic theory of annihilator immunity. This construction provides functions with the maximum possible annihilator immunity and the weight, nonlinearity and algebraic degree of the functions can be properly calculated under certain cases. The basic construction is that of symmetric Boolean functions and applying linear transformation on the input variables of these functions, one can get a large class of non-symmetric functions too. Moreover, we also study several other modifications on the basic symmetric functions to identify interesting non symmetric functions with maximum annihilator immunity. In the process we also present an algorithm to compute the Walsh spectra of a symmetric Boolean function with O(n 2 ) time and O(n) space complexity.

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“…Krawtchouk polynomial [18,20] this can be done in O(n 2 ) time and O(n) space. The basic idea is as follows:…”

mentioning

confidence: 99%

Basic Theory in Construction of Boolean Functions with Maximum Possible Annihilator Immunity

2006

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“…This conjecture was proved independently by Mignotte and Pethő [14], and by Stroeker and de Weger [16]. When n = 5, Krasikov and Litsyn [10] found a list of zeros: x N ∈ 3 17 14 36 22 67 28 67 133 289 5292 10882 . Hanrot [7] added the zero 24013 48324 to this list and showed that this new list was complete.…”

Section: Zeros Of Q-krawtchouk Polynomialsmentioning

confidence: 88%

“…When n is much larger (growing faster than √ N), Krasikov and Litsyn [10] found a better bound in the binary case:…”

Section: Zeros Of Q-krawtchouk Polynomialsmentioning

confidence: 97%

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Integer Zeros ofq-Krawtchouk Polynomials in Classical Combinatorics

Habsieger

2001

Advances in Applied Mathematics

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“…Recall that N is a positive integer, and take k 0 and k 1 such that k 1 , N), see for example [3], are defined by…”

Section: Lemmamentioning

confidence: 99%

Even moments of generalized Rudin--Shapiro polynomials

Doche

2005

Math. Comp.

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Abstract. We know from Littlewood (1968) that the moments of order 4 of the classical Rudin-Shapiro polynomials P n (z) satisfy a linear recurrence of degree 2. In a previous article, we developed a new approach, which enables us to compute exactly all the moments M q (P n ) of even order q for q 32. We were also able to check a conjecture on the asymptotic behavior of M q (P n ), namely M q (P n ) ∼ C q 2 nq/2 , where C q = 2 q/2 /(q/2 + 1), for q even and q 52. Now for every integer 2 there exists a sequence of generalized Rudin-Shapiro polynomials, denoted by P ( ) 0,n (z). In this paper, we extend our earlier method to these polynomials. In particular, the moments M q (P ( ) 0,n ) have been completely determined for = 3 and q = 4, 6, 8, 10, for = 4 and q = 4, 6 and for = 5, 6, 7, 8 and q = 4. For higher values of and q, we formulate a natural conjecture, which implies that M q (P ( ) 0,n ) ∼ C ,q nq/2 , where C ,q is an explicit constant.

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On Integral Zeros of Krawtchouk Polynomials

Abstract: We derive new conditions for the nonexistence of integral zeros of binary Krawtchouk polynomials. Upper bounds for the number of integral roots of Krawtchouk polynomials are presented.

Cited by 62 publications

References 37 publications

Basic Theory in Construction of Boolean Functions with Maximum Possible Annihilator Immunity

Basic Theory in Construction of Boolean Functions with Maximum Possible Annihilator Immunity

Integer Zeros ofq-Krawtchouk Polynomials in Classical Combinatorics

Even moments of generalized Rudin--Shapiro polynomials

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