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-<sup>k</sup>n pings) are obtained. By M (n, k) we denote , and by T (n, m, k) - the expres- s=0 у<sup>s</sup> / sion (2<sup>m</sup> - 1) <sup>П - k</sup> + M (n, k) log<sub>2</sub> . If m > 5 and k (5 + 2log<sub>2</sub>n) + 6m < < n (1/3 - y) for fixed 0 < y < 1/3, then there is n<sub>0</sub> such that, for any e<sub>1</sub> ,e<sub>2</sub> and n > n<sub>0</sub>, m<sup>2</sup> - m - 12 2 +17 M (n, k) - £<sub>1</sub> < log<sub>2</sub> |R (n, m, k)| - m2<sup>n</sup> + T (n, m, k) < < ((16m - 47) 2<sup>m 4</sup> - m + 3) M (n, k) + e<sub>2</sub>. If m > 5 and k (5 + 2log<sub>2</sub>n) + 6m < n (5/18 - y) for fixed 0 < y < 5/18, then there is n<sub>0 </sub>such that, for any e<sub>1</sub>,e<sub>2</sub> and n > n<sub>0</sub>, m<sup>2</sup> - m - 12 2 +17 M (n, k) - £<sub>1</sub> < log<sub>2</sub> |K (n, m, k) | - m2<sup>n</sup> + m2<sup>m 1</sup> + T (n, m, k) - n + 1 + log<sub>2</sub> n 2 (2<sup>m</sup> - 1) < ((16m - 47) 2<sup>m-4</sup> - m + 3) M (n, k) + £2.