Refined asymptotic estimates for the number of (n, m, k)-resilient boolean mappings

For linear combinations of coordinate functions of a random Boolean mapping, a local limit theorem for the distribution of subsets of their spectral coefficients is improved. By means of this theorem, we obtain an asymptotic formula for the |R (m, n, k)| - the number of (n, m, k)-resilient functions as n ^ ro, m G n (1 - e) / n \ G {1, 2, 3, 4} and k ^ -Ц-- for any 0 < e < 0, k = О -- : 5 + 2log₂ n Vlnnz ^log2|^R (m, n, ^k)| ~ m²ⁿ - (²m - ¹) (^ (n) +^log2 У|е(II)) + + (2 · 3^m-2 - 1) Ind {m =1} E (Л. s=0 \ ^sJ Also, we obtain upper and lower asymptotic estimates for the number |R (m, n, k)| as n ^ ro, k (5 + 2 log₂ n) + 5m ^ n (1 - e) for any 0 < e < 1: -e1 (m - 1)£ (n) < log2 |R (m,n,k)|-m2ⁿ+(2^m - ^(^(n) + log2 ^Е (n)) < < e2 (m - 2) (2^m - 1) E + E for any e^ (0 < e1,e2 < 1). s=0 \ ^s/ s=0 \ ^s/

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