On the covering radii of binary Reed-Muller codes in the set of resilient Boolean functions

Let R-t,R-n be the set of t-resilient Boolean functions in n variables, and let (p) over cap (t, r, n) be the maximum distance between t-resilient functions and the rth-order Reed-Muller code RM (r, n). We prove that,6(t, 2, 6) = 16 for t = 0, 1, 2 and)5(3, 2, 7) = 32, from which we derive the lower bound (p) over cap (t, 2, n) greater than or equal to 2(n-2) with t less than or equal to n - 4. Using a result from coding theory on the covering radius of (n - 3) th- and (n - 4)th-order Reed-Muller codes, we establish exact values of the covering radius of RM (n - 3, n) in the set of 1-resilient Boolean functions in n variables, when \n/2\ = 1 mod 2 and lower bounds of RM (n - 4, n) in the set of t-resilient Boolean functions in n variables. This result leads again to different lower bounds for general dimensions n and r = 0 or 3 mod 4.