Remarks on the generalized cyclotomic sequences of length $$2p^{m}$$

On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $ 4p^n $

“…Proof. The cases (i) and (ii) follow from Lemma 2 of [23] and Lemma 5 of [15], respectively. For the case (iii), we only prove the formula (8) since others can be proved in the same way.…”