and Circulant Hadamard Matrices

This page contains data associated with the article
Wieferich pairs and Barker sequences, II, by P. Borwein
and M. J. Mossinghoff, including lists of Wieferich prime pairs associated
with computations performed for this article, and integers *n* that
have not been eliminated as the possible length of a long Barker sequence,
or as the order of a large circulant Hadamard matrix.

Recall that a *Wieferich prime pair* (*q*, *p*) has the
property that *q*^{p−1} = 1 mod
*p*^{2}.
A *Barker sequence* is a finite sequence {*a _{i}*},
each term ±1, for which each sum of the form
∑

- The prior version of this site, containing the
data associated with the earlier article
Wieferich
primes and Barker sequences (M. J. Mossinghoff, Des. Codes
Cryptogr.
**53**(2009), no. 3, 149-163). - 156927 Wieferich prime
pairs (
*q*,*p*) with*q*<*p*required for the construction of the directed graph*D*(10^{16.5}) in the new article, but not appearing in the data listed with the earlier article. (Compressed file.) - The 4656 cycles appearing in the
directed graph
*D*(10^{16.5}), described in the latest article. - Permissible Lengths of Barker Sequences.
- Only one value of
*n*in the interval (13, 4 · 10^{33}] has not been eliminated as the possible length of a Barker sequence:*n*= 3979201339721749133016171583224100. Here,*n*= 4*u*^{2}with*u*= 31540455528264605 = 5 · 13 · 29 · 41 · 2953 · 138200401. - Nineteen integers
*u*< 5 · 10^{24}which pass all known requirements for*n*= 4*u*^{2}to be the length of a Barker sequence. - 541 integers
*u*< 5 · 10^{49}with Ω(*u*) ≤ 6 which pass all known requirements for*n*= 4*u*^{2}to be the length of a Barker sequence. - 237807 integers
*u*< 5 · 10^{49}which pass most of the requirements for*n*= 4*u*^{2}to be the length of a Barker sequence, but for which some computationally expensive tests have not been performed. (Compressed file.)

- Only one value of
- Permissible Orders of Circulant Hadamard Matrices.
- The 1371 values of
*u*≤ 10^{13}for which*n*= 4*m*^{2}has not been eliminated as the possible order of a circulant Hadamard matrix. - The 22 values of
*u*≤ 10^{13}for which*n*= 4*m*^{2}was eliminated as the possible order of a circulant Hadamard matrix by using Turyn's self-conjugacy test, but was not excluded by the computations performed in the prior paper, nor by the new methods of the 2012 article by K. H. Leung and B. Schmidt.

- The 1371 values of

- Data on
Circulant Hadamard Matrices, which contains data associated with the
article by Bernhard Schmidt and Ka Hin Leung, New restrictions on
possible orders of circulant Hadamard matrices (Des. Codes
Cryptogr.
**64**(2012), no. 1-2, 143-151). - Fermat
quotients
*q*(_{p}*a*) that are divisible by*p*, by Wilfrid Keller and Jörg Richstein. - Fermat quotients, by Richard Fischer.

Michael Mossinghoff

mimossinghoff at davidson dot edu

Last modified May 31, 2013.