Two-tuple balance of non-binary sequences with ideal two-level autocorrelation

Let p   be a prime, q=pmq=pm and FqFq be the finite field with q elements. In this paper, we will consider q  -ary sequences of period qn-1qn-1 for q>2q>2 and study their various balance properties: symbol-balance, difference-balance, and two-tuple-balance properties. The array structure of the sequences is introduced, and various implications between these balance properties and the array structure are proved. Specifically, we prove that if a q  -ary sequence of period qn-1qn-1 is difference-balanced and has the “cyclic” array structure then it is two-tuple-balanced. We conjecture that a difference-balanced q  -ary sequence of period qn-1qn-1 must have the cyclic array structure. The conjecture is confirmed with respect to all of the known q  -ary sequences which are difference-balanced, in particular, which have the ideal two-level autocorrelation function when q=pq=p.