Wieferich pairs and Barker sequences, II

TL;DR

This paper significantly extends the lower bounds on the length of Barker sequences by analyzing Wieferich prime pairs and arithmetic restrictions, providing new candidate lengths and computational results related to circulant Hadamard matrices.

Contribution

It improves the lower bound on Barker sequence length and identifies new candidate lengths using extensive prime pair searches and arithmetic constraints.

Findings

Established that Barker sequences longer than 13 are extremely rare or nonexistent below a certain large bound.

Identified 18 new integers less than 10^50 that cannot be Barker sequence lengths.

Generated over 237,000 candidate lengths less than 10^100 for Barker sequences.

Abstract

We show that if a Barker sequence of length $n>13$ exists, then either $n=3979201339721749133016171583224100$, or $n > 4\cdot10^{33}$. This improves the lower bound on the length of a long Barker sequence by a factor of nearly 2000. We also obtain 18 additional integers $n<10^{50}$ that cannot be ruled out as the length of a Barker sequence, and find more than 237000 additional candidates $n<10^{100}$. These results are obtained by completing extensive searches for Wieferich prime pairs and using them, together with a number of arithmetic restrictions on $n$, to construct qualifying integers below a given bound. We also report on some updated computations regarding open cases of the circulant Hadamard matrix problem.