Wieferich pairs and Barker sequences, II

Borwein, Peter; Mossinghoff, Michael J.

doi:10.1112/s1461157013000223

Cited by 17 publications

(12 citation statements)

References 10 publications

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“…Mossinghoff [78], Borwein and Mossinghoff [12], and Logan and Mossinghoff [71] proposed clever methods in order to identify numbers n for which Corollary 2.3.7 does not prove nonexistence of a perfect binary sequence of length n. For many of these numbers, nonexistence follows from Corollary 2.3.4 or some further nonexistence results by Leung and Schmidt [65], which also involve self-conjugacy arguments and the field descent method. Most notably, Leung and Schmidt [66] recently developed a new method, which they call the "Anti-Field-Descent Method", which provides further strong, albeit rather technical, nonexistence results.…”

Section: Nonexistence Resultsmentioning

confidence: 99%

Sequences with small correlation

Schmidt

2015

Des. Codes Cryptogr.

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The extent to which a sequence of finite length differs from a shifted version of itself is measured by its aperiodic autocorrelations. Of particular interest are sequences whose entries are 1 or −1, called binary sequences, and sequences whose entries are complex numbers of unit magnitude, called unimodular sequences. Since the 1950s, there is sustained interest in sequences with small aperiodic autocorrelations relative to the sequence length. One of the main motivations is that a sequence with small aperiodic autocorrelations is intrinsically suited for the separation of signals from noise, and therefore has natural applications in digital communications. This survey reviews the state of knowledge concerning the two central problems in this area: How small can the aperiodic autocorrelations of a binary or a unimodular sequence collectively be and how can we efficiently find the best such sequences? Since the analysis and construction of sequences with small aperiodic autocorrelations is closely tied to the (often much easier) analysis of periodic autocorrelation properties, several fundamental results on corresponding problems in the periodic setting are also reviewed.

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Section: Nonexistence Resultsmentioning

confidence: 99%

Sequences with small correlation

Schmidt

2015

Des. Codes Cryptogr.

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“…The Circulant Hadamard Conjecture [27] asserts that there does not exist any Hadamard matrix over a cyclic group Z 4t 2 , for t > 1. It is known [5,22] that this is true for all positive integer t < 11715 and also that there exist 948 open cases for t ≤ 10 13 . In the literature, this fact has dissuaded the authors from searching for cocyclic Hadamard matrices over cyclic groups.…”

Section: Cocyclic Hadamard Matricesmentioning

confidence: 93%

“…According to(5), the set S(L) is formed by the row label 1 and the symbol 2 appearing in the highlighted subarray. That is, S(L) = {1} ∪ {2} = {1, 2}.…”

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confidence: 99%

Cocyclic Hadamard matrices over Latin rectangles

Álvarez

Ganfornina

Frau

et al. 2019

European Journal of Combinatorics

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In the literature, the theory of cocyclic Hadamard matrices has always been developed over finite groups. This paper introduces the natural generalization of this theory to be developed over Latin rectangles. In this regard, once we introduce the concept of binary cocycle over a given Latin rectangle, we expose examples of Hadamard matrices that are not cocyclic over finite groups but they are over Latin rectangles. Since it is also shown that not every Hadamard matrix is cocyclic over a Latin rectangle, we focus on answering both problems of existence of Hadamard matrices that are cocyclic over a given Latin rectangle and also its reciprocal, that is, the existence of Latin rectangles over which a given Hadamard matrix is cocyclic. We prove in particular that every Latin square over which a Hadamard matrix is cocyclic must be the multiplication table of a loop (not necessarily associative). Besides, we prove the existence of cocyclic Hadamard matrices over non-associative loops of order 2 t+3 , for all positive integer t > 0.

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“…The Barker conjecture for binary sequences of even length is still open, but there is an overwhelming evidence for it. For example, results from Borwein and Mossinghoff [2] and Leung and B. Schmidt [4] show that there is no Barker sequence of even length n for 4 < n ≤ 4 • 10 33 .…”

Section: Introductionmentioning

confidence: 99%

“…So in this case only half of the aperiodic autocorrelations that are not zero are fixed; no further constraints on the other half (as for example C 1 , C √ n and hence n = 4r 2 for some r ∈ N; in particular, n is a multiple of 4. There are many more stronger and deeper partial results on the non-existence of Barker sequences of even length (see for example [3] or [2]), which however we will not use in the following.…”

Section: Introductionmentioning

confidence: 99%

A note on Barker sequences of even length

Willms

2021

Preprint

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A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are either 0, 1 or -1. The only known Barker sequences have length 2, 3, 4, 5, 7, 11 or 13. It is an old conjecture that no longer Barker sequences exist and in fact, there is an overwhelming evidence for this conjecture. For binary sequences of odd length, this conjecture is known to be true, whereas for even length it is still open, whether a Barker sequence of even length greater 4 exists.Similar to the well-known fact that a Barker sequence of odd length is necessarily skew-symmetric, we show that in the case of even length there is also a form of symmetry albeit weaker. In order to exploit this symmetry, we derive different formulas for the calculation of the aperiodic correlation. We prove by using only elementary methods that there is no Barker sequence of even length n > 4 with C, where C k denotes the kth aperiodic autocorrelation of the sequence.

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Wieferich pairs and Barker sequences, II

Cited by 17 publications

References 10 publications

Sequences with small correlation

Sequences with small correlation

Cocyclic Hadamard matrices over Latin rectangles

A note on Barker sequences of even length

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