On the Covering Radii of Binary Reed–Muller Codes in the Set of Resilient Boolean Functions
Yuri Borissov, An Braeken, Svetla Nikova⋆, Bart Preneel · IEEE Transactions on Information Theory · 2005
Let R/sub t,n/ be the set of t-resilient Boolean functions in n variables, and let /spl rho//spl circ/(t,r,n) be the maximum distance between t-resilient functions and the rth-order Reed-Muller code RM(r,n). We prove that /spl rho//spl circ/(t,2,6)=16 for t=0,1,2 and /spl rho//spl circ/(3,2,7)=32, from which we derive the lower bound /spl rho//spl circ/(t,2,n) /spl ges/ 2/sup n-2/ with t /spl les/ 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 /spl lfloor/n/2/spl rfloor/=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.