Hamming correlation of higher order

Abstract

We introduce a new measure of pseudorandomness, the (periodic) Hamming correlation of order which generalizes the Hamming autocorrelation (= 2). We analyze the relation between the Hamming correlation of order and the periodic analog of the correlation measure of order introduced by Mauduit and Sárközy. Roughly speaking, the correlation measure of order is a finer measure than the Hamming correlation of order. However, the latter can be much faster calculated and still detects some undesirable linear structures. We analyze examples of sequences with optimal Hamming correlation and show that they have large Hamming correlation of order for some very small > 2. Thus they have some undesirable linear structures, in particular in view of cryptographic applications such as secure communications.