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In mathematics, a '''±1&ndash;sequence''' is a [[sequence]] of numbers, each of which is either 1 or &minus;1. One example is the sequence {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ...)}}, where {{math|1=''x''<sub>''i''</sub> = (&minus;1)<sup>''i''+1</sup>}}.

Such sequences are commonly studied in [[discrepancy theory]].

==Erdős discrepancy problem==
<!--{{Further|[[Erdős discrepancy problem]]}}-->

Let ''S''=(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>,...) be a ±1&ndash;sequence, where ''x''<sub>''j''</sub> denotes the ''j''<sup>th</sup> term.  The ''Erdős discrepancy problem'' asks whether there exists a sequence ''S'' and an integer ''C''<sub>''S''</sub>, such that for any two positive integers ''d'' and ''k'',
: <math> \left| \sum_{i=1}^k x_{id} \right| \leq C_S.</math>
{{Asof|2010|October}}, this problem is currently being studied by the [[Polymath Project]].

==Barker codes==
{{main|Barker code}}

A '''Barker code''' is a sequence of ''N'' values of +1 and &minus;1,
:<math>a_j</math> for j = 1, 2, …, N
such that
:<math>\left|\sum_{j=1}^{N-v} a_j a_{j+v}\right| \le 1\,</math>
for all <math>1 \le v < N</math>.<ref>{{cite book |author=Barker, R. H. |chapter=Group Synchronizing of Binary Digital Sequences |title=Communication Theory |place=London |publisher=Butterworth |pages=273–287 |year=1953}}</ref>

Barker codes of length 11 and 13 are used in [[direct-sequence spread spectrum]]  and [[pulse compression|pulse compression radar]] systems because of their low [[autocorrelation]] properties.

==See also==
*[[Binary sequence]]
*[[Discrepancy of hypergraphs]]

==Notes==
{{reflist}}

==References==
*{{cite book|title=The Discrepancy Method: Randomness and Complexity|last=Chazelle|first=Bernard|author-link=Bernard Chazelle|publisher=Cambridge University Press|ISBN=0-521-77093-9}}

==External links==
*[http://michaelnielsen.org/polymath1/index.php?title=The_Erd%C5%91s_discrepancy_problem The Erdős discrepancy problem] &ndash; Polymath Project

{{DEFAULTSORT:±1-Sequence}}
[[Category:Binary sequences]]