A further study on the linear complexity of new binary cyclotomic sequence of length $p^r$

Recently, a conjecture on the linear complexity of a new class of generalized cyclotomic binary sequences of period $p^r$ was 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 $r\ge 1$, Vladimir Edemskiy proved the conjecture (arXiv:1712.03947). In this paper, under the assumption of $2^{p-1} \not\equiv 1 \bmod p^2$ and $\gcd(\frac{p-1}{{\rm {ord}}_{p}(2)},f)=1$, the conjecture proposed by Z. Xiao et al. is proved for a general $f$ by using the Euler quotient. Actually, a generic construction of $p^r$-periodic binary sequence based on the generalized cyclotomy is introduced in this paper, which admits a flexible support set and includes Xiao's construction as a special case, and then an efficient method to compute the linear complexity of the sequence by the generic construction is presented, based on which the conjecture proposed by Z. Xiao et al. could be easily proved under the aforementioned assumption.