Classification, construction and search of general quasi-orthogonal binary signal sets

In this paper, we derive a kind of classification of binary signal set by adopting Hadamard equivalence of binary matrices. To do this, we study various properties of this equivalence relation and its classes. We propose to use a concept named “HR-minimal” as a representative of each equivalence class, and some properties and constructions of HR-minimals are investigated. Especially, careful observation about the weight on an HR-minimal´s second row are performed since it is highly related with orthogonality of a matrix. We give a construction ensuring sufficiently large m at some condition. We also give an exhaustive search algorithm to find the maximum m at given n and second row weight. Moreover, HR-minimals with the largest weight on its second row, defined as Quasi-Hadamard matrices (QH matrices), are studied. They include Hadamard matrices as a special case and generalize the property of Hadamard matrices in the sense that the row vectors of m × n QH matrices form a set of m binary vectors of length n with minimum pairwise absolute correlation. Some properties and existence of QH matrices are also discussed, including the examples of 16 × (6, 10, 17) QH matrices found by computer.