Two-circulant Hadamard matrices, weighing matrices, and Ryser’s conjecture
Keywords:
Information Processing, Noise-Immune Coding, Masking Images, Orthogonal Matrices, Quasi-Orthogonal Hadamard Matrices, Belevitch Matrices, Weighing Matrices, Two-Circulant Matrices, Ryser's ConjectureAbstract
Hadamard matrices and weighing matrices share the same family. The latter can fill up voids in the matrix space by setting some elements to zero, but this feature has not been properly studied yet. Purpose: To study how the orders of orthogonal matrices used in information processing can affect their structure. Results: For Ryser's conjecture about orders critical for cyclic Hadamard matrices, an extension has been suggested, covering Hadamard matrices and weighing matrices which consist of two cyclic blocks. We give examples of Hadamard matrices extended to the newly revealed critical order equal to 32, with symmetrical blocks or, on higher orders, with unsymmetrical blocks. We also present two-circulant weighing matrices which replace Hadamard matrices and alternate with them. There is an exceptional case related to the order 24 on which two-circulant Hadamard matrices or weighing matrices do not exist, forcing you to search for a solution among four-block constructions. A special set of Hadamard matrices of 20- and 52-fold orders is pointed out, as their blocks are asymmetric. A new assumption about the critical order 64 is discussed.