A Proof of Turyn's Conjecture: Nonexistence of Circulant Hadamard Matrices for Order Greater than Four

Yoshimasa OH-HASHI  

Publication
IEICE TRANSACTIONS on Communications   Vol.E99-B   No.7   pp.1395-1407
Publication Date: 2016/07/01
Online ISSN: 1745-1345
DOI: 10.1587/transcom.2015EBP3506
Type of Manuscript: PAPER
Category: Fundamental Theories for Communications
Keyword: 
finite field,  biphase periodic sequence,  two-level autocorrelation,  ring isomorphism,  circulant Hadamard matrix,  Chinese remainder theorem,  

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Summary: 
Biphase periodic sequences having elements +1 or -1 with the two-level autocorrelation function are desirable in communications and radars. However, in case of the biphase orthogonal periodic sequences, Turyn has conjectured that there exist only sequences with period 4, i.e., there exist the circulant Hadamard matrices for order 4 only. In this paper, it is described that the conjecture is proved to be true by means of the isomorphic mapping, the Chinese remainder theorem, the linear algebra, etc.