Bisecting binomial coefficients

Ionascu, Eugen J.; Martinsen, Thor; Stănică, Pantelimon

doi:10.1016/j.dam.2017.04.035

Cited by 5 publications

(10 citation statements)

References 18 publications

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“…Table 1 contains γ j (n) for various n's and j's. [13]. As part of their work, they consider the number of nontrivial bisections.…”

Section: Diophantine Equations With Binomial Coefficientsmentioning

confidence: 99%

“…Some of these results may be extended to perturbations of symmetric Boolean functions. For example, following similar techniques form [13] we obtained an integral representation for γ j (n). Let V j = [0, 2 j−1 ] ∩ Z.…”

Section: Diophantine Equations With Binomial Coefficientsmentioning

confidence: 99%

“…There are only four of such perturbations of degree less than or equal to 14. The first one is σ 15,[5,6,10,12,13] + X 1 , which corresponds to the equivalent solution The third one is σ 15,[6,7,11,12,14] + X 1 and it corresponds to the equivalent solution…”

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confidence: 99%

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Diophantine Equations With Binomial Coefficients and Perturbations of Symmetric Boolean Functions

Castro

González

Medina

2018

IEEE Trans. Inform. Theory

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Abstract. This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form To be specific, it is proved that, excluding the trivial cases, balanced perturbations of fixed degree do not exist when the number of variables grows. Some sporadic balanced perturbations are presented. Finally, a beautiful but unexpected identity between perturbations of two very different symmetric Boolean functions is also included in this work.

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“…Table 1 contains γ j (n) for various n's and j's. [13]. As part of their work, they consider the number of nontrivial bisections.…”

Section: Diophantine Equations With Binomial Coefficientsmentioning

confidence: 99%

Section: Diophantine Equations With Binomial Coefficientsmentioning

confidence: 99%

See 1 more Smart Citation

Diophantine Equations With Binomial Coefficients and Perturbations of Symmetric Boolean Functions

Castro

González

Medina

2018

IEEE Trans. Inform. Theory

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“…is said to give a bisection of the binomial coefficients n j , 0 ≤ j ≤ n. The problem of bisecting binomial coefficients is an interesting problem in its own right, however, it is out of the scope of this work. The interested reader is invited to read [14,15].…”

Section: Preliminariesmentioning

confidence: 99%

Closed formulas for exponential sums of symmetric polynomials over Galois fields

2018

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Exponential sums have applications to a variety of scientific fields, including, but not limited to, cryptography, coding theory and information theory. Closed formulas for exponential sums of symmetric Boolean functions were found by Cai, Green and Thierauf in the late 1990's. Their closed formulas imply that these exponential sums are linear recursive. The linear recursivity of these sums has been exploited in numerous papers and has been used to compute the asymptotic behavior of such sequences. In this article, we extend the result of Cai, Green and Thierauf, that is, we find closed formulas for exponential sums of symmetric polynomials over any Galois fields. Our result also implies that the recursive nature of these sequences is not unique to the binary field, as they are also linear recursive over any finite field. In fact, we provide explicit linear recurrences with integer coefficients for such sequences. As a byproduct of our results, we discover a link between exponential sums of symmetric polynomials over Galois fields and a problem for multinomial coefficients which similar to the problem of bisecting binomial coefficients.

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“…Cusick and Li ([3]) raised some questions about the set of values n so that only the trivial solutions of (BCBP) exist. Theorem 2.6 from [5], was rediscovered in [4], and this provided a positive answer to questions Q2 in Q4 ( [3], page 86). Question Q1 is still open, and we want to include it explicitly: Q1: Are there infinitely many odd values of n for which only the trivial solutions of (BCBP) exist?…”

Section: Introductionmentioning

confidence: 99%