A new restriction on the lengths of golay complementary sequences

Eliahou, Shalom; Kervaire, Michel; Saffari, Bahman

doi:10.1016/0097-3165(90)90046-y

Cited by 80 publications

(51 citation statements)

References 5 publications

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“…2D and 4C], [9], [10]). In 1990, Eliahou, Kervaire, and Saffari [11] proved that if p | m then p ≡ 1 (mod 4); in 1992 Eliahou and Kervaire [9] and Jedwab and Lloyd [19] both used this constraint, together with some additional restrictions on m, to show that there are no Barker sequences with 1 < m < 689. In 1999, Schmidt [30] obtained much stronger restrictions on m, determining that no Barker sequences exist with m ≤ 10 6 .…”

Section: Barker Sequencesmentioning

confidence: 99%

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: Barker Sequencesmentioning

confidence: 99%

Barker sequences and flat polynomials

Borwein¹,

Mossinghoff²

2008

Number Theory and Polynomials

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“…On the nonexistence side we have the following two results; the original proof of Theorem 4 in [9] was elegantly shortened in [10]: Proposition 3 (Golay [16]). If there exists a binary Golay sequence pair of length s > 1 then s is even.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 4 (Eliahou, Kervaire and Saffari [9], [10]). If there exists a binary Golay sequence pair of length s then s has no prime factor congruent to 3 modulo 4.…”

Section: Introductionmentioning

confidence: 99%

Golay complementary array pairs

Jedwab

Parker

2007

Des. Codes Cryptogr.

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“…Turyn and Storer [18] proved that no Barker sequences of odd length exist for n 13. Various restrictions on the possible values of n were derived in papers [4], [5], [8], [12], [16]. It is known that if a Barker sequence of even length n > 13 exists, then either n = 189 260 468 001 034 441 522 766 781 604, or n > 2 · 10 30 by the result of [9], though the nonexistence problem still remains unsolved and has been for more than 45 years.…”

Section: Introductionmentioning

confidence: 99%

On a class of polynomials related to Barker sequences

Borwein

Choi

Jankauskas

2012

Proc. Amer. Math. Soc.

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Abstract. For an odd integer n > 0, we introduce the class LP n of Laurent polynomialswith all coefficients c k equal to −1 or 1. Such polynomials arise in the study of Barker sequences of even length, i.e., integer sequences having minimal possible autocorrelations. We prove that polynomials P ∈ LP n have large Mahler measures, namely, M (P ) > (n + 1)/2. We conjecture that minimal Mahler measures in the class LP n are attained by the polynomials R n (z) and R n (−z), whereis a polynomial with all the coefficients c k = 1. We prove thatThe results of experimental computations on polynomials in the class LP n suggest two conjectures which could shed light on the long-standing Barker problem.

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A new restriction on the lengths of golay complementary sequences

Cited by 80 publications

References 5 publications

Barker sequences and flat polynomials

Barker sequences and flat polynomials

Golay complementary array pairs

On a class of polynomials related to Barker sequences

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