Run Vector Analysis and Barker Sequences of Odd Length

TL;DR

This paper investigates the run vector of skew-symmetric binary sequences and provides a new proof that Barker sequences of odd length greater than 13 do not exist, enhancing understanding of sequence autocorrelations.

Contribution

It introduces a novel analysis of the run vector for skew-symmetric sequences and offers a new proof regarding the nonexistence of long odd-length Barker sequences.

Findings

No Barker sequences of odd length n > 13 exist.

Established a strong relationship between run vectors and autocorrelations.

Provided a new proof method for Barker sequence nonexistence.

Abstract

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.