Polyphase Barker sequences up to length 36

Polyphase sequences are finite, complex, time-discrete sequences with constant magnitude and variable phases. A polyphase sequence with elements of magnitude 1 is called a Barker sequence if the maximum magnitude of all sidelobes of their aperiodic auto-correlation function (ACF) is less or equal to one. Stochastic optimization algorithms have been applied to search for polyphase Barker sequences. New sequences meeting the Barker condition up to length 31 have been found. Introduction: Polyphase sequences with low autocorrelation sidelobes are applied for example in radar and system identification. Because of the constant magnitude they have a valuable feature: With a given peak power constraint on the transmitter their energy is higher than that of any other non-constant magnitude sequence. The low magnitude of the ACF sidelobes in case of Barker sequences ensures an easily detectable peak at the output of a matched filter receiver. Binary Barker sequences with elements a n ∈ {−1, +1} are only known up to length 13 [1]. Increasing the size of the alphabet allows the construction of longer Barker sequences. M-Polyphase sequences have elements a n ∈ {exp(2πi/M)} with i = 0. .. M − 1. They can be regarded as a generalization of binary sequences which are included as a special case with M = 2. When M gets infinite, the phases become continuous variables and the sequence is called uniform. This paper refers to polyphase sequences with large M .