Improved estimates for the number of (n, m, k)-resilient and correlation-immune boolean mappings

Improved lower and upper bounds for |K (n, m, k)| (the number of correlation-immune of order k binary mappings) and |R (m,n, k)| (the number of (n, m, k)-resilient binary map-^kn pings) are obtained. By M (n, k) we denote , and by T (n, m, k) - the expres- s=0 у^s / sion (2^m - 1) ^{П - k} + M (n, k) log₂ . If m > 5 and k (5 + 2log₂n) + 6m < < n (1/3 - y) for fixed 0 < y < 1/3, then there is n₀ such that, for any e₁ ,e₂ and n > n₀, m² - m - 12 2 +17 M (n, k) - £₁ < log₂ |R (n, m, k)| - m2ⁿ + T (n, m, k) < < ((16m - 47) 2^{m 4} - m + 3) M (n, k) + e₂. If m > 5 and k (5 + 2log₂n) + 6m < n (5/18 - y) for fixed 0 < y < 5/18, then there is n₀ such that, for any e₁,e₂ and n > n₀, m² - m - 12 2 +17 M (n, k) - £₁ < log₂ |K (n, m, k) | - m2ⁿ + m2^{m 1} + T (n, m, k) - n + 1 + log₂ n 2 (2^m - 1) < ((16m - 47) 2^m-4 - m + 3) M (n, k) + £2.

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