# ±1-sequence

In mathematics, a ±1–sequence is a sequence of numbers, each of which is either 1 or −1. One example is the sequence (x1, x2, x3, ...), where xi = (−1)i+1.

Such sequences are commonly studied in discrepancy theory.

## Erdős discrepancy problem

Let S=(x1, x2, x3,...) be a ±1–sequence, where xj denotes the jth term. The Erdős discrepancy problem asks whether there exists a sequence S and an integer CS, such that for any two positive integers d and k,

$\left| \sum_{i=1}^k x_{id} \right| \leq C_S.$

As of October 2010, this problem is currently being studied by the Polymath Project.

## Barker codes

A Barker code is a sequence of N values of +1 and −1,

$a_j$ for j = 1, 2, …, N

such that

$\left|\sum_{j=1}^{N-v} a_j a_{j+v}\right| \le 1\,$

for all $1 \le v < N$.[1]

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