A NOTE ON BARKER POLYNOMIALS
A NOTE ON BARKER POLYNOMIALS
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关于 Barker 多项式的注记
DOI：
10.1142/s179304211250159x
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发表时间：
2012
影响因子：
0.7
通讯作者：
T. Erdélyi
中科院分区：
文献类型：
--
作者：
P. Borwein;T. Erdélyi
如果每个aj∈{- 1,1}，我们称该多项式为n-1次的Barker多项式。Barker多项式的性质在60年代初由Turyn和stover进行了深入的研究，在80年代末由Saffari进行了研究。在过去的几年里，P. Borwein和他的合作者重新燃起了对巴克多项式（巴克码，巴克序列）研究的兴趣。本文给出了一个新的证明，证明不存在大于12的偶数次的Barker多项式，因而不存在大于13的奇长Barker序列。这与不可约性问题密切相关，并由以下新结果证明。定理。如果n对j的偶数点来说，每个bj∈{- 1,0,1}，对j的奇数点来说，每个bj是一个能被4整除的整数，则不存在这样的多项式，其中，表示系数在{- 1,1}的所有2m次多项式的集合。牛顿恒等式的巧妙运用在我们优雅的证明中起着核心作用。
We call the polynomial a Barker polynomial of degree n-1 if each aj ∈{-1, 1} and Properties of Barker polynomials were studied by Turyn and Storer thoroughly in the early sixties, and by Saffari in the late eighties. In the last few years P. Borwein and his collaborators revived interest in the study of Barker polynomials (Barker codes, Barker sequences). In this paper we give a new proof of the fact that there is no Barker polynomial of even degree greater than 12, and hence Barker sequences of odd length greater than 13 do not exist. This is intimately tied to irreducibility questions and proved as a consequence of the following new result. Theorem.Ifn ≔ 2m + 1 > 13andwhere eachbj ∈{-1, 0, 1}for even values of j, each bj is an integer divisible by 4 for odd values of j, then there is no polynomialsuch that, whereanddenotes the collection of all polynomials of degree 2m with each of their coefficients in {-1, 1}. A clever usage of Newton's identities plays a central role in our elegant proof.