arXiv subjects

Jürgen Willms

Publications and source records attributed to Jürgen Willms.

5 recordsLinked to original sources

arXiv2021

A note on Barker sequences of even length

A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are either 0, 1 or -1. The only known Barker sequences have length 2, 3, 4, 5, 7, 11 or 13. It is an old conjecture that no longer Barker sequences exist and in fact, there is an overwhelming evidence for this conjecture. For binary sequences of odd length, this conjecture is known to be true, whereas for even length it is still open, whether a Barker sequence of even length greater 4 exists. Similar to the well-known fact that a Barker sequence of odd length is necessarily skew-symmetric, we show that in the case of even length there is also a form of symmetry albeit weaker. In order to exploit this symmetry, we derive different formulas for the calculation of the aperiodic correlation. We prove by using only elementary methods that there is no Barker sequence of even length n>4 with $C_{1}=C_{3}=\cdots=C_{\frac{n}{2}-1}$, where $C_{k}$ denotes the $k$th aperiodic autocorrelation of the sequence.

math.CO↗

arXiv2015

Barker sequences of odd length

A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are at most 1 in magnitude. An old conjecture due to Turyn asserts that there is no Barker sequence of length greater than 13. In 1961, Turyn and Storer gave an elementary, though somewhat complicated, proof that this conjecture holds for odd lengths. We give a new and simpler proof of this result.

math.CO↗

arXiv2014

Run Vector Analysis and Barker Sequences of Odd Length

The run vector of a binary sequence reflects the run structure of the sequence, which is given by the set of all substrings of the run length encoding. The run vector and the aperiodic autocorrelations of a binary sequence are strongly related. In this paper, we analyze the run vector of skew-symmetric binary sequences. Using the derived results we present a new and different proof that there exists no Barker sequence of odd length n > 13. Barker sequences are binary sequences whose off-peak aperiodic autocorrelations are all in magnitude at most 1.

cs.IT↗

arXiv2014

Counterexamples to Theorem 1 of Turyn's and Storer's Paper "On Binary Sequences"

Explicit counterexamples to Theorem 1 of R. Turyn's and J. Storer's often-cited paper "On Binary Sequences" [Proc. Amer. Math. Soc., vol. 12, no. 3, pp. 394-399, 1961] are given. Theorem 2 of their paper is well known; it states that Barker sequences of odd length n>13 do not exist. Since the proof of Theorem 2 relies on the incorrect Theorem 1, the proof of the often-cited Theorem 2 is therefore not correct. It is not at all clear how Theorem 1 and its proof must be modified in order to get a correct proof of the well-known Theorem 2.

math.NT↗

arXiv2013

Autocorrelations of Binary Sequences and Run Structure

We analyze the connection between the autocorrelation of a binary sequence and its run structure given by the run length encoding. We show that both the periodic and the aperiodic autocorrelation of a binary sequence can be formulated in terms of the run structure. The run structure is given by the consecutive runs of the sequence. Let C=(C(0), C(1),...,C(n)) denote the autocorrelation vector of a binary sequence. We prove that the kth component of the second order difference operator of C can be directly calculated by using the consecutive runs of total length k. In particular this shows that the kth autocorrelation is already determined by all consecutive runs of total length L<k. In the aperiodic case we show how the run vector R can be efficiently calculated and give a characterization of skew-symmetric sequences in terms of their run length encoding.

cs.IT↗