# A further study of the linear complexity of new binary cyclotomic sequence of length \(p^r\)

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## Abstract

Recently, a conjecture on the linear complexity of a new class of generalized cyclotomic binary sequences of period \(p^r\) was proposed by Xiao et al. (Des Codes Cryptogr 86(7):1483–1497, 2018). Later, for the case *f* being the form \(2^a\) with \(a\ge 1\), 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 \(\gcd (\frac{p-1}{\mathrm{{ord}}_{p}(2)},f)=1\), the conjecture proposed by Xiao et al. is proved for a general *f* by using the Euler quotient. Actually, a generic construction of \(p^r\)-periodic binary sequences based on the generalized cyclotomy is introduced in this paper, which admits a flexible support set and contains 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 Xiao et al. could be easily proved under the aforementioned assumption.

## Keywords

Linear complexity Generalized cyclotomy Binary sequence Euler quotient## Notes

### Acknowledgements

The authors would like to thank the reviewers and editors for their detailed and constructive comments, which substantially improved the presentation of the paper. This work was supported by the National Natural Science Foundation of China (No. 61772292, 61772476), Foundation of Fujian Educational Committee (No. JAT170627), Key Scientific Research Projects of Colleges and Universities in Henan Province (No. 18A110029) and Fujian Normal University Innovative Research Team (No. IRTL1207).

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