Parameterization of Perfect Sequences over a Composition Algebra

Takao MAEDA  Takafumi HAYASHI  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E95-A   No.12   pp.2139-2147
Publication Date: 2012/12/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E95.A.2139
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Sequence
parameterization,  perfect sequence,  autocorrelation,  composition algebra,  quaternion,  octonion,  

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A parameterization of perfect sequences over composition algebras over the real number field is presented. According to the proposed parameterization theorem, a perfect sequence can be represented as a sum of trigonometric functions and points on a unit sphere of the algebra. Because of the non-commutativity of the multiplication, there are two definitions of perfect sequences, but the equivalence of the definitions is easily shown using the theorem. A composition sequence of sequences is introduced. Despite the non-associativity, the proposed theorem reveals that the composition sequence from perfect sequences is perfect.