Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle
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Fedor M. Malyshev
Abstract
The paper is concerned with estimating the number ξ of ones in triangular arrays consisting of elements of the field GF(2) which are defined by the bottom row of s elements. The elements of each higher row are obtained (as in Pascal triangles) by the summation of pairs of elements from the corresponding lower row. It is shown that there exists a monotone unbounded sequence 0 = k0 < k1 < k2 < ... of rational numbers such that, for any k > 0, for sufficiently large s the admissible values of ξ which are smaller than ks or larger than s(s + 1)/3 − sk/3 are concentrated in neighbourhoods of points kis and s(s + 1)/3 − ski/3, i ⩾ 0. The resulting estimates of the neighbourhoods are functions of i for each i ⩾ 0 and do not depend on s. The distributions of the numbers of triangles with values ξ in these neighbourhoods depend only on the residues of s with respect to moduli that depend on i ⩾ 0.
Originally published in Diskretnaya Matematika (2016) 28, №3, 59-96 (in Russian).
References
[1] Malyshev F. M., “Basises of integers under the multiplace shift operations”, Mat. Vopr. Kriptogr., 2:1 (2011), 29-74 (in Russian).10.4213/mvk25Search in Google Scholar
[2] Wolfram S., “Cellular automaton supercomputing”, In: High-Speed Computing, Univ. of Illinois Press, 1988,40-48.10.1201/9780429494093-19Search in Google Scholar
[3] Malyshev F. M., Kutyreva E. V., “On the distribution of the number of ones in a Boolean Pascal’s triangle”, Discrete Math. Appl., 16:3 (2006), 271-279.10.1515/156939206777970435Search in Google Scholar
[4] Malyshev F. M., “Basises of recurrent sequences”, Chebyshevskiy sbornik, 16:2 (2015), 155-185 (in Russian).Search in Google Scholar
[5] Berlekamp E. R., Algebraic Coding Theory, McGraw Hill, 1968.Search in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Bounds for the average-case complexity of monotone Boolean functions
- Modular algorithm for reducing matrices to the Smith normal form
- Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle
- Application of Hadamard product to some combinatorial and probabilistic problems
- Cardinality of subsets of the residue group with nonunit differences of elements
Articles in the same Issue
- Frontmatter
- Bounds for the average-case complexity of monotone Boolean functions
- Modular algorithm for reducing matrices to the Smith normal form
- Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle
- Application of Hadamard product to some combinatorial and probabilistic problems
- Cardinality of subsets of the residue group with nonunit differences of elements
