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

Let R_t,n be the set of t-resilient Boolean functions in n variables, and let ρˆ(t,r,n) be the maximum distance between t-resilient functions and the rth-order Reed-Muller code RM(r,n). We prove that ρˆ(t,2,6)=16 for t=0,1,2 and ρˆ(3,2,7)=32, from which we derive the lower bound ρˆ(t,2,n) ≥ 2^n-2 with t ≤ 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 2-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.