Advances in Mathematics of Communications最新文献

Aperodic (or called Golay)/Periodic complementary pairs (GCPs/ PCPs) are pairs of sequences whose aperiodic/periodic autocorrelation sums are zero everywhere, except at the zero shift. In this paper, we introduce GCPs/PCPs over the quaternion group begin{document}$ Q_8 $end{document}, which is a generalization of quaternary GCPs/PCPs. Some basic properties of autocorrelations of begin{document}$ Q_8 $end{document}-sequences are also obtained. We present three types of constructions for GCPs and PCPs over begin{document}$ Q_8 $end{document}. The main ideas of these constructions are to consider pairs of a begin{document}$ Q_8 $end{document}-sequence and its reverse, pairs of interleaving of sequence, or pairs of Kronecker product of sequences. By choosing suitable sequences in these constructions, we obtain new parameters for GCPs and PCPs, which have not been reported before.

Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting subject of study in recent years. The construction of optimal cyclic codes over finite fields is important as they have maximal minimum distance once the length and dimension are given. In this paper, we present two classes of new optimal ternary cyclic codes begin{document}$ mathcal{C}_{(2,v)} $end{document} by using monomials begin{document}$ x^2 $end{document} and begin{document}$ x^v $end{document} for some suitable begin{document}$ v $end{document} and explain the novelty of the codes. Furthermore, the weight distribution of begin{document}$ mathcal{C}_{(2,v)}^{perp} $end{document} for begin{document}$ v = frac{3^{m}-1}{2}+2(3^{k}+1) $end{document} is determined.

We present a proof under a generalization of the Riemann Hypothesis that the class group algorithm of Hafner and McCurley runs in expected time begin{document}$ e^{left(3/sqrt{8}+o(1)right)sqrt{log dloglog d}} $end{document} where begin{document}$ -d $end{document} is the discriminant of the input imaginary quadratic order. In the original paper, an expected run time of begin{document}$ e^{left(sqrt{2}+o(1)right)sqrt{log dloglog d}} $end{document} was proven, and better bounds were conjectured. To achieve a proven result, we rely on a mild modification of the original algorithm, and on recent results on the properties of the Cayley graph of the ideal class group.

This paper mainly study begin{document}$ mathbb{Z}_{2}mathbb{Z}_{4}[u] $end{document}-additive codes. A Gray map from begin{document}$ mathbb{Z}_{2}^{alpha}timesmathbb{Z}_{4}^{beta}[u] $end{document} to begin{document}$ mathbb{Z}_{4}^{alpha+2beta} $end{document} is defined, and we prove that is a weight preserving and distance preserving map. A MacWilliams-type identity between the Lee weight enumerator of a begin{document}$ mathbb{Z}_{2}mathbb{Z}_{4}[u] $end{document}-additive code and its dual is proved. Some properties of one-weight begin{document}$ mathbb{Z}_{2}mathbb{Z}_{4}[u] $end{document}-additive codes and two-weight projective begin{document}$ mathbb{Z}_{2}mathbb{Z}_{4}[u] $end{document}-additive codes are discussed. As main results, some construction methods for one-weight and two-weight begin{document}$ mathbb{Z}_{2}mathbb{Z}_{4}[u] $end{document}-additive codes are studied, meanwhile several examples are presented to illustrate the methods.