On the symmetry of finite pseudorandom binary sequences (Q2893511)

\textit{K. Gyarmati} [Ramanujan J. 8, 289--302 (2004; Zbl 1062.11053)] introduced a new measure of pseudorandomness: the symmetry measure for binary sequences \(E_N=\{e_1,\dots ,e_N\}\in \{-1,1\}^N\). Using the idea of multiple symmetry centers and arithmetically symmetric subsequences the author introduces two new measures of pseudo-randomness which eliminate symmetrical patterns. Lower bounds are proved for both measures holding for sufficiently long binary sequences. It is also shown that certain upper bounds hold for majority of sequences. Finally a sequence is shown which passes the usual statistical tests for small correlation, well-distribution and symmetry measures, but has deeply symmetric structure.