Abstract
We present a generalised setting for the construction of complementary array pairs and its proof, using a unitary matrix notation. When the unitaries comprise multivariate polynomials in complex space, we show that four definitions of conjugation imply four types of complementary pair - types I, II, III, and IV. We provide a construction for complementary pairs of types I, II, and III over {1, − 1}, and further specialize to a construction for all known 2 ×2 ×…×2 complementary array pairs of types I, II, and III over {1, − 1}. We present a construction for type-IV complementary array pairs, and call them Rayleigh quotient pairs. We then generalise to complementary array sets, provide a construction for complementary sets of types I, II, and III over {1, − 1}, further specialize to a construction for all known 2 ×2 ×…×2 complementary array sets of types I, II, and III over {1, − 1}, and derive closed-form Boolean formulas for these cases.
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