When the necessary conditions are not sufficient: sequences with zero autocorrelation function

Recently K. T. Arasu (personal communication) and Yoseph Strassler, in his PhD thesis, The Classification of Circulant Weighing Matrices of Weight 9, Bar-Ilan University, Ramat-Gan, 1997, have intensively studied circulant weighing matrices, or single sequences, with weight 9. They show many cases are non-existent. Here we give details of a search for two sequences with zero periodic autocorrelation and types (1,9), (1,16) and (4,9). We find some new cases but also many cases where the known necessary conditions are not sufficient. We instance a number of occasions when the known necessary conditions are not sufficient for the existence of weighing matrices and orthogonal de-signs constructed using sequences with zero autocorrelation function leading to intriguing new questions

Multiplication of ternary complementary pairs

We present a computer-search method for concatenating or multiplying binary or ternary complementary pairs. All multiplications by a particular number m are considered. The computer-search method is new and leads to a large set of new results. The results and equivalences are discussed and some applications and numerical consequences are shown

Constructions for orthogonal designs using signed group orthogonal designs

Craigen introduced and studied signed group Hadamard matrices extensively and eventually provided an asymptotic existence result for Hadamard matrices. Following his lead, Ghaderpour introduced signed group orthogonal designs and showed an asymptotic existence result for orthogonal designs and consequently Hadamard matrices. In this paper, we construct some interesting families of orthogonal designs using signed group orthogonal designs to show the capability of signed group orthogonal designs in generation of different types of orthogonal designs.Comment: To appear in Discrete Mathematics (Elsevier). No figure

On sufficient conditions for some orthogonal designs and sequences with zero autocorrelation function

We give new sets of sequences with entries from {0, ±a, ±b, ±c, ±d} on the commuting variables a, b, c, d and zero autocorrelation function. Then we use these sequences to construct some new orthogonal designs. This means that for order 28 only the existence of the following five cases, none of which is ruled out by known theoretical results, remain in doubt: OD(28; 1, 4, 9, 9), OD(28; 1, 8, 8, 9), OD(28; 2, 8, 9, 9), OD(28; 3, 6, 8, 9), OD(28; 4, 4, 4, 9). We consider 4 - N PAF(Sl, S2, S3, S4) sequences or four sequences of commuting variables from the set {0, ±a, ±b, ±c, ±d} with zero nonperiodic autocorrelation function where ±a occurs Sl times, ±b occurs S2 times, etc. We show the necessary conditions for the existence of an 0D(4n; S1,S2, S3,S4) constructed using four circulant matrices are sufficient conditions for the existence of 4 - NPAF(S1, S2, S3, S4) sequences for all lengths ≥ n, i) for n = 3, with the extra condition (S1,S2,S3,S4) ≠ (1,1,1,9), ii) for n = 5, provided there is an integer matrix P satisfying PPT = diag (S1,S2,S3,S4), iii) for n = 7, with the extra condition that (S1,S2,S3,S4) ≠ (1,1,1,25), and possibly (S1,S2,S3,S4) =I- (1,1,1,16), (1,1,8,18), (1,1,13,13), (1,4,4,9), (1,4,9,9), (1,4,10,10), (1,8,8,9), (1,9,9,9), (2,4,4,18), (2,8,9,9), (3,4,6,8), (3,6,8,9); (4,4,4,9), (4,4,9,9), (4,5,5,9), (5,5,9,9). We show the necessary conditions for the existence of an OD(4n; S1,S2) constructed using four circulant matrices are sufficient conditions for the existence of 4 - NPAF(S1,S2) sequences for all lengths ≥ n, where n = 3 or 5

Fast Decoder for Overloaded Uniquely Decodable Synchronous Optical CDMA

In this paper, we propose a fast decoder algorithm for uniquely decodable (errorless) code sets for overloaded synchronous optical code-division multiple-access (O-CDMA) systems. The proposed decoder is designed in a such a way that the users can uniquely recover the information bits with a very simple decoder, which uses only a few comparisons. Compared to maximum-likelihood (ML) decoder, which has a high computational complexity for even moderate code lengths, the proposed decoder has much lower computational complexity. Simulation results in terms of bit error rate (BER) demonstrate that the performance of the proposed decoder for a given BER requires only 1-2 dB higher signal-to-noise ratio (SNR) than the ML decoder.Comment: arXiv admin note: substantial text overlap with arXiv:1806.0395

Formation of regulatory modules by local sequence duplication

Turnover of regulatory sequence and function is an important part of molecular evolution. But what are the modes of sequence evolution leading to rapid formation and loss of regulatory sites? Here, we show that a large fraction of neighboring transcription factor binding sites in the fly genome have formed from a common sequence origin by local duplications. This mode of evolution is found to produce regulatory information: duplications can seed new sites in the neighborhood of existing sites. Duplicate seeds evolve subsequently by point mutations, often towards binding a different factor than their ancestral neighbor sites. These results are based on a statistical analysis of 346 cis-regulatory modules in the Drosophila melanogaster genome, and a comparison set of intergenic regulatory sequence in Saccharomyces cerevisiae. In fly regulatory modules, pairs of binding sites show significantly enhanced sequence similarity up to distances of about 50 bp. We analyze these data in terms of an evolutionary model with two distinct modes of site formation: (i) evolution from independent sequence origin and (ii) divergent evolution following duplication of a common ancestor sequence. Our results suggest that pervasive formation of binding sites by local sequence duplications distinguishes the complex regulatory architecture of higher eukaryotes from the simpler architecture of unicellular organisms

Detection of signals by weighted integrate-and-dump filter

A Weighted Integrate and Dump Filter (WIDF) is presented that results in reducing those losses in telemetry symbol signal to noise ratio (SNR) which occur in digital Integrate and Dump Filters (IDFs) when the samples are not phase locked to the input data symbol clock. The Minimum Mean Square Error (MMSE) criterion is used to derive a set of weights for approximating the analog integrate and dump filter, which is the matched filter for detection of signals in additive white Gaussian noise. This new digital matched filter results in considerable performance improvement compared to unweighted digital matched filters. An example is presented for a sampling rate of four times the symbol rate. As the sampling offset (or phase) varies with respect to the data symbol boundaries, the output SNR varies 1 dB for an unweighted IDF, but only 0.3 dB for the optimum WIDF, averaged over random data patterns. This improvement in performance relative to unweighted IDF means that significantly lower sampling and processing rates can be used for given telemetry symbol rates, resulting in reduced system cost