A note on a conjecture of new binary cyclotomic sequences of length p^n

12/24/2017

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by   Zhifan Ye, et al.

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Recently, a conjecture on the linear complexity of a new class of generalized cyclotomic binary sequences of period p^n were proposed by Z. Xiao et al. (Des. Codes Cryptogr. DOI 10.1007/s10623-017-0408-7). Later, for the case f being the form 2^r with r> 1, Vladimir Edemskiy proved the conjecture (arXiv:1712.03947). In this paper, we first introduce a generic construction of p^n-periodic binary sequence based on the generalized cyclotomy, which admits a flexible support set and includes Xiao's construction as a special case. Then, under the assumption of 2 being a primitive root modulo p^2, the linear complexity of the new proposed sequence over GF (2) is determined by using the Euler quotient. As a byproduct, in the case of 2 being a primitive root modulo p^2, the conjecture given by Z. Xiao et al. is proved to be correct for a general f.