Merit factors of polynomials derived from difference sets

Günther, Christian; Schmidt, Kai-Uwe

doi:10.1016/j.jcta.2016.08.006

Cited by 11 publications

(8 citation statements)

References 18 publications

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“…One can also crosscorrelate modified Legendre sequences h s,ℓ p with f s,ℓ p and g s,ℓ p and obtain crosscorrelation performance on par with random sequences while maintaining high autocorrelation performance. We should remark that the autocorrelation behavior of f s,ℓ p and g s,ℓ p was independently discovered by Günther and Schmidt and reported in [10]. They obtain the same formulae for asymptotic autocorrelation merit factor as presented here in Theorem 8.3.…”

Section: Introductionsupporting

confidence: 75%

Low Correlation Sequences From Linear Combinations of Characters

Boothby

Katz

2017

IEEE Trans. Inform. Theory

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Abstract. Pairs of binary sequences formed using linear combinations of multiplicative characters of finite fields are exhibited that, when compared to random sequence pairs, simultaneously achieve significantly lower mean square autocorrelation values (for each sequence in the pair) and significantly lower mean square crosscorrelation values. If we define crosscorrelation merit factor analogously to the usual merit factor for autocorrelation, and if we define demerit factor as the reciprocal of merit factor, then randomly selected binary sequence pairs are known to have an average crosscorrelation demerit factor of 1. Our constructions provide sequence pairs with crosscorrelation demerit factor significantly less than 1, and at the same time, the autocorrelation demerit factors of the individual sequences can also be made significantly less than 1 (which also indicates better than average performance). The sequence pairs studied here provide combinations of autocorrelation and crosscorrelation performance that are not achievable using sequences formed from single characters, such as maximal linear recursive sequences (msequences) and Legendre sequences. In this study, exact asymptotic formulae are proved for the autocorrelation and crosscorrelation merit factors of sequence pairs formed using linear combinations of multiplicative characters. Data is presented that shows that the asymptotic behavior is closely approximated by sequences of modest length.

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Section: Introductionsupporting

confidence: 75%

Low Correlation Sequences From Linear Combinations of Characters

Boothby

Katz

2017

IEEE Trans. Inform. Theory

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“…The challenge of determining the maximum MF for binary sequences is an old and complicated combinatorial optimization problem [51]. In communication engineering and statistical mechanics such as the Bernasconi model [52], the main problem is the mathematically Littlewood polynomials [53].…”

Section: Review Of the State-of-the-art Work Of Labs Techniquesmentioning

confidence: 99%

A Hybrid Modified Sine Cosine Algorithm Using Inverse Filtering and Clipping Methods for Low Autocorrelation Binary Sequences

Rosli¹,

Rahim²,

Rani³

et al. 2022

CMC

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The essential purpose of radar is to detect a target of interest and provide information concerning the target's location, motion, size, and other parameters. The knowledge about the pulse trains' properties shows that a class of signals is mainly well suited to digital processing of increasing practical importance. A low autocorrelation binary sequence (LABS) is a complex combinatorial problem. The main problems of LABS are low Merit Factor (MF) and shorter length sequences. Besides, the maximum possible MF equals 12.3248 as infinity length is unable to be achieved. Therefore, this study implemented two techniques to propose a new metaheuristic algorithm based on Hybrid Modified Sine Cosine Algorithm with Cuckoo Search Algorithm (HMSCACSA) using Inverse Filtering (IF) and clipping method to achieve better results. The proposed algorithms, LABS-IF and HMSCACSA-IF, achieved better results with two large MFs equal to 12.12 and 12.6678 for lengths 231 and 237, respectively, where the optimal solutions belong to the skew-symmetric sequences. The MF outperformed up to 24.335% and 2.708% against the state-of-the-art LABS heuristic algorithm, xLastovka, and Golay, respectively. These results indicated that the proposed algorithm's simulation had quality solutions in terms of fast convergence curve with better optimal means, and standard deviation.

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“…for sequences derived from linear combinations of multiplicative characters, which includes the binary cyclotomic sequences as a proper subclass. Boothby and Katz's conditions are met by the cyclotomic sequences derived from quartic characters described at (16) and (17), and these sequences were used by both Boothby and Katz [2,Theorem 19] and Günther and Schmidt [16, p. 347] as examples showing that one can achieve asymptotic demerit factor 0.157 . .…”

Section: Sequences From Multiplicative Charactersmentioning

confidence: 99%

“…Boothby and Katz [2, Theorem 21] crosscorrelated the two cyclotomic sequences (16) and (17) derived from quartic characters, and also the cyclically shifted versions of these two sequences. As mentioned in Section 7, one obtains very low asymptotic autocorrelation demerit factor only for certain lengths, depending on a number-theoretic criterion.…”

Section: Pairs With Low Asymptotic Pursley-sarwate Criterionmentioning

confidence: 99%

Sequences with Low Correlation

Katz

2018

Arithmetic of Finite Fields

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Pseudorandom sequences are used extensively in communications and remote sensing. Correlation provides one measure of pseudorandomness, and low correlation is an important factor determining the performance of digital sequences in applications. We consider the problem of constructing pairs (f, g) of sequences such that both f and g have low mean square autocorrelation and f and g have low mean square mutual crosscorrelation. We focus on aperiodic correlation of binary sequences, and review recent contributions along with some historical context.

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Merit factors of polynomials derived from difference sets

Cited by 11 publications

References 18 publications

Low Correlation Sequences From Linear Combinations of Characters

Low Correlation Sequences From Linear Combinations of Characters

A Hybrid Modified Sine Cosine Algorithm Using Inverse Filtering and Clipping Methods for Low Autocorrelation Binary Sequences

Sequences with Low Correlation

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