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On the Nonexistence of Almost Difference Sets Constructed from the Set of Octic Residues
Minglong QI Shengwu XIONG Jingling YUAN Wenbi RAO Luo ZHONG
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E99A
No.2
pp.666673 Publication Date: 2016/02/01 Online ISSN: 17451337
DOI: 10.1587/transfun.E99.A.666 Type of Manuscript: LETTER Category: Cryptography and Information Security Keyword: binary sequence, threelevel autocorrelation, difference set, almost difference set, set of octic residues,
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Summary:
Pseudorandom binary sequences balanced and with optimal autocorrelation have many applications in the stream cipher, communication, coding theory, etc. Constructing a binary sequences with threelevel autocorrelation is equivalent to finding the corresponding characteristic set of the sequences that should be an almost difference set. In the work of T.W. Cusick, C. Ding, and A. Renvall in 1998, the authors gave the necessary and sufficient conditions by which a set of octic residues modulo an odd prime forms an almost difference set. In this paper we show that no integers verify those conditions by the theory of generalized Pell equations. In addition, by relaxing the definition of almost difference set given by the same authors, we could construct two classes of modified almost difference sets and two ones of difference sets from the set of octic residues.

