Symmetric measures of pseudorandomness for binary sequences

We compare ordinary and symmetric variants of two classical measures of pseudorandomness for binary sequences, the $2$-adic complexity and the linear complexity. In the periodic setting, we show that for binary periodic sequences constructed from the binary expansions of non-palindromic primes, the symmetric $2$-adic complexity can be strictly smaller than the ordinary $2$-adic complexity. We also give a direct proof (of the known result) that the linear complexity of a periodic binary sequence is invariant under reversal, and hence coincides with its symmetric version. In the aperiodic setting, we provide explicit families of finite binary sequences for which both the $N$th symmetric 2-adic complexity and the $N$th symmetric linear complexity are substantially smaller than their ordinary counterparts. Furthermore, we show that the expected values of the $N$th rational complexity and of the $N$th exponential linear complexity exceed those of their symmetric analogues by at least a term of order of magnitude $N$. Thus, the effect of symmetrization is clearly visible on an exponential scale. We also establish lower bounds for the expected values of the symmetric rational complexity, symmetric $2$-adic complexity, symmetric linear complexity, and symmetric exponential linear complexity.