New restrictions on possible orders of circulant Hadamard matrices

Abstract

We obtain several new number theoretic results which improve the field descent method. We use these results to rule out many of the known open cases of the circulant Hadamard matrix conjecture. In particular, the only known open case of the Barker sequence conjecture is settled.

Douglas-Rachford feasibility methods for matrix completion problems

“…The circulant Hadamard conjecture asserts: No circulant Hadamard matrix of order larger than 4 exists. For recent progress on the conjecture, see [37]. Consistent with this conjecture, our Douglas-Rachford implementation can successfully find circulant matrices of order 4, but fails for higher orders.…”

Wieferich pairs and Barker sequences, II

“…In [5], Leung and Schmidt obtained another restriction that rules out values of n having a sizable prime-power divisor, provided that a side condition also holds. For a prime p and integer t, let ν p (t) denote the largest integer k such that p k | t.…”

“…In [5], Leung and Schmidt established a second bound depending on the same function F (m, n). We cite their result here only as it applies to circulant Hadamard matrices.…”

“…The same article established that fewer than 1600 integers n < 4 · 10 26 satisfy all the known restrictions in the circulant Hadamard matrix problem. Recently, Leung and Schmidt [5] obtained some further restrictions on the order of a circulant Hadamard matrix, and some of these restrictions apply to the Barker sequence problem as well. These new restrictions are also summarized in § 2.…”

“…Our method follows that of [6], but incorporates the new restrictions of [5]. This method relies on a large search for Wieferich prime pairs, which are pairs of prime numbers (q, p) with the property that q p−1 ≡ 1 mod p 2 .…”