Periodic Golay pairs and pairwise balanced designs

In this paper we exploit a relationship between certain pairwise balanced designs with v points and periodic Golay pairs of length v, to classify periodic Golay pairs of length less than 40. In particular, we construct all pairwise balanced designs with v points under specific block conditions having an assumed cyclic automorphism group, and using isomorph rejection which is compatible with equivalence of corresponding periodic Golay pairs, we complete a classification up to equivalence. This is done using the theory of orbit matrices and some compression techniques which apply to complementary sequences. We use similar tools to construct new periodic Golay pairs of lengths greater than 40 where classifications remain incomplete and demonstrate that under some extra conditions on its automorphism group, a periodic Golay pair of length 90 will not exist. Length 90 remains the smallest length for which existence of a periodic Golay pair is undecided. Some quasi-cyclic self-orthogonal codes are constructed as an added application

Evidence of Intermittent Cascades from Discrete Hierarchical Dissipation in Turbulence

We present the results of a search of log-periodic corrections to scaling in the moments of the energy dissipation rate in experiments at high Reynolds number (2500) of three-dimensional fully developed turbulence. A simple dynamical representation of the Richardson-Kolmogorov cartoon of a cascade shows that standard averaging techniques erase by their very construction the possible existence of log-periodic corrections to scaling associated with a discrete hierarchy. To remedy this drawback, we introduce a novel ``canonical'' averaging that we test extensively on synthetic examples constructed to mimick the interplay between a weak log-periodic component and rather strong multiplicative and phase noises. Our extensive tests confirm the remarkable observation of statistically significant log-periodic corrections to scaling, with a prefered scaling ratio for length scales compatible with the value gamma = 2. A strong confirmation of this result is provided by the identification of up to 5 harmonics of the fundamental log-periodic undulations, associated with up to 5 levels of the underlying hierarchical dynamical structure. A natural interpretation of our results is that the Richardson-Kolmogorov mental picture of a cascade becomes a realistic description if one allows for intermittent births and deaths of discrete cascades at varying scales.Comment: Latex document of 40 pages, including 18 eps figure

Large Zero Autocorrelation Zone of Golay Sequences and 4q4^q-QAM Golay Complementary Sequences

Sequences with good correlation properties have been widely adopted in modern communications, radar and sonar applications. In this paper, we present our new findings on some constructions of single $H$-ary Golay sequence and $4^q$-QAM Golay complementary sequence with a large zero autocorrelation zone, where $H\ge 2$ is an arbitrary even integer and $q\ge 2$ is an arbitrary integer. Those new results on Golay sequences and QAM Golay complementary sequences can be explored during synchronization and detection at the receiver end and thus improve the performance of the communication system

New Evidence of Discrete Scale Invariance in the Energy Dissipation of Three-Dimensional Turbulence: Correlation Approach and Direct Spectral Detection

We extend the analysis of [Zhou and Sornette, Physica D 165, 94-125, 2002] showing statistically significant log-periodic corrections to scaling in the moments of the energy dissipation rate in experiments at high Reynolds number ($\approx 2500$) of three-dimensional fully developed turbulence. First, we develop a simple variant of the canonical averaging method using a rephasing scheme between different samples based on pairwise correlations that confirms Zhou and Sornette's previous results. The second analysis uses a simpler local spectral approach and then performs averages over many local spectra. This yields stronger evidence of the existence of underlying log-periodic undulations, with the detection of more than 20 harmonics of a fundamental logarithmic frequency $f = 1.434 \pm 0.007$ corresponding to the preferred scaling ratio $\gamma = 2.008 \pm 0.006$.Comment: 9 RevTex4 papes including 8 eps figure