Des. Codes Cryptogr. (2012) 62:209–223 DOI 10.1007/s10623-011-9506-0 Intersection of Hamming codes avoiding Hamming subcodes J. Rifà · F. I. Solov’eva · M. Villanueva Received: 20 December 2010 / Revised: 30 March 2011 / Accepted: 31 March 2011 / Published online: 17 April 2011 © Springer Science+Business Media, LLC 2011 Abstract We prove that given a binary Hamming code H n of length n = 2 m - 1, m ≥ 3, or equivalently a projective geometry PG(m - 1, 2), there exist permutations π ∈ S n , such that H n and π(H n ) do not have any Hamming subcode with the same support, or equiva- lently the corresponding projective geometries do not have any common flat. The introduced permutations are called AF permutations. We study some properties of these permutations and their relation with the well known APN functions. Keywords APN functions · Cryptography · Hamming codes · Intersection of Hamming codes · Projective geometries Mathematics Subject Classification (2000) 94B25 · 05B25 · 14G50 1 Introduction Let F n be the vector space of dimension n over the binary field F. The Hamming distance between vectors x , y ∈ F n , denoted by d (x , y ), is the number of coordinates in which x and y differ. The Hamming weight of x ∈ F n , denoted by w(x ), is given by w(x ) = d (x , 0), where 0 is the all-zero vector. The support of a vector x ∈ F n is the set of nonzero coordinate positions of x and is denoted by supp(x ). Given a binary code C and a subcode C ′ of C, the Communicated by S. Ball. J. Rifà (B ) · M. Villanueva Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain e-mail: josep.rifa@autonoma.edu M. Villanueva e-mail: merce.villanueva@autonoma.edu F. I. Solov’eva Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia e-mail: sol@math.nsc.ru 123