Abstract
We say a polynomial p ( z ) = a n z n + a n − 1 z n − 1 + ⋯ + a 0 is a Littlewood polynomial if a k = ± 1 for 0 ⩽ k ⩽ n . Let p ( z ) p ( 1 / z ) = c n z n + c n − 1 z n − 1 + ⋯ + c − n z − n . It is easy to show that c 0 = n + 1 . We say that p ( z ) is a Barker polynomial if | c k | ⩽ 1 for k ≠ 0 . There are only 8 known Barker polynomials (normalized to have a n = a n − 1 = 1 ). There are many results known about the existence and non-existence of Barker polynomials for various degrees. This paper deals with the infinite case, when f ( z ) = ± 1 ± z ± z 2 ± ⋯ is a power series with ±1 coefficients. We give a complete description of all Barker series.
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