On binary sequences

Turyn, R.; Storer, James A.

doi:10.1090/s0002-9939-1961-0125026-2

Cited by 127 publications

(43 citation statements)

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“…As we noted in section 2, Turyn and Storer [35] proved that no Barker sequences of odd length n exist for n > 13. Their proof is elementary, though somewhat complicated, and relies on showing that long Barker sequences of odd length must exhibit certain patterns.…”

Section: An Irreducibility Questionmentioning

confidence: 77%

“…The following results are due to Turyn and Storer [35,32]; we include the proof here for the reader's convenience.…”

Section: Barker Sequencesmentioning

confidence: 97%

“…Turyn and Storer [35] proved that if the length n of a Barker sequence is odd then n ≤ 13, so the complete list for this case appears in Table 1. It also follows from this that no additional sequences satisfy Barker's original requirement for sequences whose off-peak autocorrelations are all 0 or −1, since Theorem 2.1 implies that any such sequence must have length n ≡ 3 (mod 4).…”

Section: Barker Sequencesmentioning

confidence: 99%

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Barker sequences and flat polynomials

Borwein¹,

Mossinghoff²

2008

Number Theory and Polynomials

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Abstract. A Barker sequence is a finite sequence of integers, each ±1, whose aperiodic autocorrelations are all as small as possible. It is widely conjectured that only finitely many Barker sequences exist. We describe connections between Barker sequences and several problems in analysis regarding the existence of polynomials with ±1 coefficients that remain flat over the unit circle according to some criterion. First, we amend an argument of Saffari to show that a polynomial constructed from a Barker sequence remains within a constant factor of its L 2 norm over the unit circle, in connection with a problem of Littlewood. Second, we show that a Barker sequence produces a polynomial with very large Mahler's measure, in connection with a question of Mahler. Third, we optimize an argument of Newman to prove that any polynomial with ±1 coefficients and positive degree n − 1 has L 1 norm less than √ n − .09, and note that a slightly stronger statement would imply that long Barker sequences do not exist. We also record polynomials with ±1 coefficients having maximal L 1 norm or maximal Mahler's measure for each fixed degree up to 24. Finally, we show that if one could establish that the polynomials in a particular sequence are all irreducible over Q, then an alternative proof that there are no long Barker sequences with odd length would follow.

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Section: An Irreducibility Questionmentioning

confidence: 77%

“…The following results are due to Turyn and Storer [35,32]; we include the proof here for the reader's convenience.…”

Section: Barker Sequencesmentioning

confidence: 97%

Section: Barker Sequencesmentioning

confidence: 99%

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Barker sequences and flat polynomials

Borwein¹,

Mossinghoff²

2008

Number Theory and Polynomials

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“…We give a short alternative proof to that in [16]. For a binary sequence A of length s, it is well-known (see, for example, [25]) that…”

Section: Introductionmentioning

confidence: 95%

Golay complementary array pairs

Jedwab

Parker

2007

Des. Codes Cryptogr.

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“…The best achievable with a binary codes is an autocorrelation with a sharp central peak and remaining elements, or "sidelobes", having peak size 1; the codes that achieve this are the wellknown Barker codes [1] [6]. However, odd-length Barker codes exist only up to length 13 [12], and for even-length Barkers, only up to length 4, if the well-known "Barker Conjecture" holds true [13]. One reason why it is impossible to do better is that the size of the outermost sidelobes is 1, since it is the product of the size of two unit-magnitude code elements.…”

Section: Introductionmentioning

confidence: 99%

Efficient exhaustive search for binary complementary code sets

Coxson

Russo

2013

2013 47th Annual Conference on Information Sciences and Systems (CISS)

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Abstract-Binary complementary code sets offer a possibility that single binary codes cannot -zero aperiodic autocorrelation sidelobe levels. These code sets can be viewed as the columns of so-called complementary code matrices, or CCMs. This matrix formulation is particularly useful in gaining the insight needed for developing an efficient exhaustive search for complementary code sets. An exhaustive search approach is described, designed to find all sets of K complementary binary codes of length N , for specified N and K. Results for several cases are examined.

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On binary sequences

Cited by 127 publications

References 1 publication

Barker sequences and flat polynomials

Barker sequences and flat polynomials

Golay complementary array pairs

Efficient exhaustive search for binary complementary code sets

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