论文标题
有限领域的不确定性原则
The uncertainty principle over finite fields
论文作者
论文摘要
在本文中，我们研究了不确定性原理（向上）在有限场上连接一个函数及其Mattson-Solomon多项式，这是一种阳性特征的一种傅立叶变换。研究了三个版本的有限领域，与循环代码的渐近理论有关。我们首先表明，没有有限的字段满足EVRA，KOWALSKY，LUBOTZKY，2017年最近推出的强大版本。通过使用渐近plotkin Bound Kind，给出了弱版本的改进。通过使用BCH绑定，证明了一个天真的版本，它是Donoho-Stark绑定的有限磁场上的直接类似物。足够强，可以表明，所有$ 0 <α<1/2 $的循环代码序列有长度$ n $，任意速率和最小距离$ω（n^α）$的序列。最后，指出了与拉姆齐理论的联系。
In this paper we study the uncertainty principle (UP) connecting a function over a finite field and its Mattson-Solomon polynomial, which is a kind of Fourier transform in positive characteristic. Three versions of the UP over finite fields are studied, in connection with the asymptotic theory of cyclic codes. We first show that no finite field satisfies the strong version of UP, introduced recently by Evra, Kowalsky, Lubotzky, 2017. A refinement of the weak version is given, by using the asymptotic Plotkin bound. A naive version, which is the direct analogue over finite fields of the Donoho-Stark bound over the complex numbers, is proved by using the BCH bound. It is strong enough to show that there exist sequences of cyclic codes of length $n$, arbitrary rate, and minimum distance $Ω(n^α)$ for all $0<α<1/2$. Finally, a connection with Ramsey Theory is pointed out.
