Cyclic Difference Sets And Cyclic Hadamard Matrices

TL;DR

This paper proves that only two cyclic Hadamard matrices exist, leading to the conclusion that there are exactly seven Barker sequences over {-1, 1}, resolving a longstanding open problem.

Contribution

The paper establishes a complete classification of cyclic Hadamard matrices, showing only two exist, and deduces the exact number of Barker sequences over {-1, 1}.

Findings

Only two cyclic Hadamard matrices exist.

Exactly seven Barker sequences over {-1, 1} are possible.

The result resolves an open problem in combinatorial design theory.

Abstract

The collection of cyclic Hadamard matrices {H = (a_{i - j}) : 0 <= i, j < n, and a_i = -1, 1} of order n is characterized by the orthogonality relation HH^T = nI. Only two of such matrices are currently known. It will be shown that this collection consists of precisely two matrices. An application of this result implies that there are exactly seven Barker sequences over the binary set {-1, 1}.