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 $\Omega(n^\alpha)$ for all $0<\alpha<1/2$. Finally, a connection with Ramsey Theory is pointed out.
翻译:在本文中,我们研究了不确定性原则(UP),该原则将有限字段的功能与其马特森-索洛蒙多元海洋学相联系,这是一种富丽雅的正性变异,结合循环编码的零时论理论,研究了三个版本的优于有限字段。我们首先表明,没有一个限定字段能够满足最近Evra、Kowalsky、Lubotzky, 2017年引进的强值UP的版本。通过使用无药方图盘,对薄弱版本进行了改进。一个天真版本,即对多诺霍-Stark的有限字段进行直接比对复杂数字的比喻,通过使用BCH约束来证明,它足够强大,足以表明存在长度为$、任意率和最低距离为$Omega(n ⁇ alpha)$ <1/2美元(n ⁇ alpha)的周期代码序列。最后指出,与Ramsey Theory的连接点是。