A Simple Proof for the Existence of Exponentially Balanced Gray Codes A Gray code of length n is a circular list of all 2n bitstrings or binary codewords of length n such that successive codewords differ in only one bit position. The frequencies of the positions where these differences occur are called transition counts. An exponentially balanced Gray code is a Gray code the transition counts of which are all the same power of two, or are two successive powers of two. A proof for the existence of exponentially balanced Gray codes is derived. The proof is much simpler than an earlier proof presented by van Zanten and Suparta (Discrete Analysis and Operation Research, 11 (2004) 8198 (Russian Journal)).

New Bounds on 2Frameproof Codes of Length 4 Frameproof codes were first introduced by Boneh and Shaw in 1998 in the context of digital fingerprinting to protect copyrighted materials. These digital fingerprints are generally denoted as codewords in          ,  where Q is an alphabet of size q and n is a positive integer. A 2frameproof code is a code C such that any 2 codewords in C cannot form a new codeword under a particular rule. Thus, no pair of users can frame a user who is not a member of the coalition. This paper concentrates on the upper bound for the size of a qary 2frameproof code of length 4. Our new upper bound shows that                                        when q is odd and                .