26,212 research outputs found

    Sequences with Low Correlation

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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)(f,g) of sequences such that both ff and gg have low mean square autocorrelation and ff and gg have low mean square mutual crosscorrelation. We focus on aperiodic correlation of binary sequences, and review recent contributions along with some historical context.Comment: 24 page

    Advances in the merit factor problem for binary sequences

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    The identification of binary sequences with large merit factor (small mean-squared aperiodic autocorrelation) is an old problem of complex analysis and combinatorial optimization, with practical importance in digital communications engineering and condensed matter physics. We establish the asymptotic merit factor of several families of binary sequences and thereby prove various conjectures, explain numerical evidence presented by other authors, and bring together within a single framework results previously appearing in scattered form. We exhibit, for the first time, families of skew-symmetric sequences whose asymptotic merit factor is as large as the best known value (an algebraic number greater than 6.34) for all binary sequences; this is interesting in light of Golay's conjecture that the subclass of skew-symmetric sequences has asymptotically optimal merit factor. Our methods combine Fourier analysis, estimation of character sums, and estimation of the number of lattice points in polyhedra.Comment: 31 pages, minor revision

    Note on the Merit Factors of Sequences

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    A method for estimating the merit factors of sequences will be provided. The result is also effective in determining the nonexistence of certain infinite collections of cyclic difference sets and cyclic matrices and associated binary sequences.Comment: Key Words: Binary sequences, Merit factor, Barker sequence, cyclic Hadamard matrix, Menon difference sets. 19 page

    Modifications on Character Sequences and Construction of Large Even Length Binary Sequences

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    It has been noticed that all the known binary sequences having the asymptotic merit factor ≥6\ge 6 are the modifications to the real primitive characters. In this paper, we give a new modification of the character sequences at length N=p1p2…prN=p_1p_2\dots p_r, where pip_i's are distinct odd primes and rr is finite. Based on these new modifications, for N=p1p2…prN=p_1p_2\dots p_r with pip_i's distinct odd primes, we can construct a binary sequence of length 2N2N with asymptotic merit factor $6.0

    Low Correlation Sequences from Linear Combinations of Characters

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    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 11. Our constructions provide sequence pairs with crosscorrelation demerit factor significantly less than 11, and at the same time, the autocorrelation demerit factors of the individual sequences can also be made significantly less than 11 (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 (m-sequences) 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.Comment: 47 page

    Aperiodic Crosscorrelation of Sequences Derived from Characters

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    It is shown that pairs of maximal linear recursive sequences (m-sequences) typically have mean square aperiodic crosscorrelation on par with that of random sequences, but that if one takes a pair of m-sequences where one is the reverse of the other, and shifts them appropriately, one can get significantly lower mean square aperiodic crosscorrelation. Sequence pairs with even lower mean square aperiodic crosscorrelation are constructed by taking a Legendre sequence, cyclically shifting it, and then cutting it (approximately) in half and using the halves as the sequences of the pair. In some of these constructions, the mean square aperiodic crosscorrelation can be lowered further if one truncates or periodically extends (appends) the sequences. Exact asymptotic formulae for mean squared aperiodic crosscorrelation are proved for sequences derived from additive characters (including m-sequences and modified versions thereof) and multiplicative characters (including Legendre sequences and their relatives). Data is presented that shows that sequences of modest length have performance that closely approximates the asymptotic formulae.Comment: 54 page

    Rudin-Shapiro-Like Sequences with Maximum Asymptotic Merit Factor

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    Borwein and Mossinghoff investigated the Rudin-Shapiro-like sequences, which are infinite families of binary sequences, usually represented as polynomials. Each family of Rudin-Shapiro-like sequences is obtained from a starting sequence (which we call the seed) by a recursive construction that doubles the length of the sequence at each step, and many sequences produced in this manner have exceptionally low aperiodic autocorrelation. Borwein and Mossinghoff showed that the asymptotic autocorrelation merit factor for any such family is at most 33, and found the seeds of length 4040 or less that produce the maximum asymptotic merit factor of 33. The definition of Rudin-Shapiro-like sequences was generalized by Katz, Lee, and Trunov to include sequences with arbitrary complex coefficients, among which are families of low autocorrelation polyphase sequences. Katz, Lee, and Trunov proved that the maximum asymptotic merit factor is also 33 for this larger class. Here we show that a family of such Rudin-Shapiro-like sequences achieves asymptotic merit factor 33 if and only if the seed is either of length 11 or is the interleaving of a pair of Golay complementary sequences. For small seed lengths where this is not possible, the optimal seeds are interleavings of pairs that are as close as possible to being complementary pairs, and the idea of an almost-complementary pair makes sense of remarkable patterns in previously unexplained data on optimal seeds for binary Rudin-Shapiro-like sequences.Comment: 22 page

    Crosscorrelation of Rudin-Shapiro-Like Polynomials

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    We consider the class of Rudin-Shapiro-like polynomials, whose L4L^4 norms on the complex unit circle were studied by Borwein and Mossinghoff. The polynomial f(z)=f0+f1z+⋯+fdzdf(z)=f_0+f_1 z + \cdots + f_d z^d is identified with the sequence (f0,f1,…,fd)(f_0,f_1,\ldots,f_d) of its coefficients. From the L4L^4 norm of a polynomial, one can easily calculate the autocorrelation merit factor of its associated sequence, and conversely. In this paper, we study the crosscorrelation properties of pairs of sequences associated to Rudin-Shapiro-like polynomials. We find an explicit formula for the crosscorrelation merit factor. A computer search is then used to find pairs of Rudin-Shapiro-like polynomials whose autocorrelation and crosscorrelation merit factors are simultaneously high. Pursley and Sarwate proved a bound that limits how good this combined autocorrelation and crosscorrelation performance can be. We find infinite families of polynomials whose performance approaches quite close to this fundamental limit.Comment: 32 page

    Family complexity and cross-correlation measure for families of binary sequences

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    We study the relationship between two measures of pseudorandomness for families of binary sequences: family complexity and cross-correlation measure introduced by Ahlswede et al.\ in 2003 and recently by Gyarmati et al., respectively. More precisely, we estimate the family complexity of a family (ei,1,…,ei,N)∈{−1,+1}N(e_{i,1},\ldots,e_{i,N})\in \{-1,+1\}^N, i=1,…,Fi=1,\ldots,F, of binary sequences of length NN in terms of the cross-correlation measure of its dual family (e1,n,…,eF,n)∈{−1,+1}F(e_{1,n},\ldots,e_{F,n})\in \{-1,+1\}^F, n=1,…,Nn=1,\ldots,N. We apply this result to the family of sequences of Legendre symbols with irreducible quadratic polynomials modulo pp with middle coefficient 00, that is, ei,n=(n2−bi2p)n=1(p−1)/2e_{i,n}=\left(\frac{n^2-bi^2}{p}\right)_{n=1}^{(p-1)/2} for i=1,…,(p−1)/2i=1,\ldots,(p-1)/2, where bb is a quadratic nonresidue modulo pp, showing that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order

    Merit factors of polynomials derived from difference sets

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    The problem of constructing polynomials with all coefficients 11 or −1-1 and large merit factor (equivalently with small L4L^4 norm on the unit circle) arises naturally in complex analysis, condensed matter physics, and digital communications engineering. Most known constructions arise (sometimes in a subtle way) from difference sets, in particular from Paley and Singer difference sets. We consider the asymptotic merit factor of polynomials constructed from other difference sets, providing the first essentially new examples since 1991. In particular we prove a general theorem on the asymptotic merit factor of polynomials arising from cyclotomy, which includes results on Hall and Paley difference sets as special cases. In addition, we establish the asymptotic merit factor of polynomials derived from Gordon-Mills-Welch difference sets and Sidelnikov almost difference sets, proving two recent conjectures.Comment: 22 pages, this revision contains a more general version of Thm. 2.
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