In this paper we present a construction method for quaternary sequences from a binary sequence of even period, which preserves the period and autocorrelation of the given binary sequence. By applying the method to the binary sequences with three-valued autocorrelation, we construct new quaternary sequences with three-valued autocorrelation, which are balanced or almost balanced. In particular, we construct new balanced quaternary sequences whose autocorrelations are three-valued and have out-of-phase magnitude 2, when their periods are N=pm-1 and N≡ 2 (mod 4) for any odd prime p and any odd integer m. Their out-of-phase autocorrelation magnitude is the known optimal value for N≠ 2,4,8, and 16.

The linear complexity and its stability of periodic sequences are of fundamental importance as measure indexes on the security of stream ciphers and the k-error linear complexity reveals the stability of the linear complexity properly. The k-error linear complexity of periodic sequences is defined to be the smallest linear complexity that can be obtained by changing k or fewer bits of the sequence per period. For 2pn-periodic binary sequences, where p is an odd prime and 2 is a primitive root modulo p2, we present and prove the unique expression of the linear complexity. Moreover we show a relationship between the linear complexity and the minimum value k for which the k-error linear complexity is strictly less than the linear complexity.

It is known that homogeneous networks are ones which perform parallel algorithms, and the dynamics of neural networks are applied to practical problems including combinatorial optimization problems. Both homogeneous and neural networks are parallel networks, and are composed of Boolean elements. Although a large number of studies have been made on the applications of homogeneous threshold networks, little is known about the relation of the dynamics of these networks. In this paper, some results about the dynamics, used to find the lengths of periodic and transient sequences, as built by parallel networks including threshold and homogeneous networks are shown. First, we will show that for non–restricted parallel networks, threshold networks which permit only two elements to transit at each step, and homogeneous networks, it is possible to build periodic and transient sequences of almost any lengths. Further, it will be shown that it is possible for triangular threshold networks to build periodic and transient sequences with short lengths only. As well, homogeneous threshold networks also seem to build periodic and transient sequences with short lengths only. Specifically, we will show a sufficient condition for symmetric homogeneous threshold networks to have periodic sequences with the length 1.