Symmetry of Two-Circulant Hadamard Matrices and Periodic Golay Pairs
Abstract
Purpose: Construction of Hadamard matrices of two-circulant type. The role of symmetry and skew-symmetry of the circulant blocks in this construction is investigated systematically. Another goal of this work is to classify the periodic Golay pairs up to length 40. These pairs are closely related to the above mentioned construction. Methods: Computational methods of linear algebra, recursive methods of optimum search, exhaustive search to construct all periodic Golay pairs of a fixed length by using high-performance computers. Results: The paper discusses the problem of constructing Hadamard matrices of two-circulant type by introducing certain measures of symmetry (symmetry index, defects of symmetry and skew-symmetry) and enumerates the equivalence classes of periodic Golay pairs of small lengths. An analog of the Ryser's conjecture, the non-existence of circulant Hadamard matrices of order bigger than 4, has been proposed earlier by the first author. It asserts that there are no symmetric Hadamard matrices of two-circulant type and of order bigger than 32. The latter conjecture is verified in several cases by using a computer. A catalogue of the representatives of equivalence classes of two-circulant Hadamard matrices is presented in the form of a list of periodic Golay pairs of lengths up to 26 (inclusive). Examples of nearly symmetric two-circulant Hadamard matrices of relatively large order are given. Practical relevance: Hadamard matrices have direct practical applications to the problems of noise-immune coding and compression and masking of video information. Software for constructing two-circulant Hadamard matrices and a library of periodic Golay pairs, together with the online algorithms, are made available on the mathematical network http://mathscinet.ru.Published
2015-06-01
How to Cite
Balonin, N., & Djokovic, D. (2015). Symmetry of Two-Circulant Hadamard Matrices and Periodic Golay Pairs. Information and Control Systems, (3), 2-16. https://doi.org/10.15217/issn1684-8853.2015.3.2
Issue
Section
Theoretical and applied mathematics