Binary Sequence Pairs with Two-Level Correlation and Cyclic Difference Pairs

Abstract

We investigate binary sequence pairs with two-level correlation in terms of their corresponding cyclic difference pairs (CDPs). We define multipliers of a cyclic difference pair and present an existence theorem for multipliers, which could be applied to check the existence/nonexistence of certain hypothetical cyclic difference pairs. Then, we focus on the ideal case where all the out-of-phase correlation coefficients are zero. It is known that such an ideal binary sequence pair exists for length v= 4u for every u ≥ 1. Using the techniques developed here on the theory of multipliers of a CDP and some exhaustive search, we are able to determine that, for lengths v ≤ 30, (1) there does not exist “any other” ideal binary sequence pair and (2) every example in this range is equivalent to the one of length v = 4u above. We conjecture that if there is a binary sequence pair with an ideal two-level correlation then its in-phase correlation must be 4. This implies so called the circulant Hadamard matrix conjecture.