论文标题
关于魏尔总和，猜想的希勒斯和NIHO指数
On Weil Sums, Conjectures of Helleseth, and Niho Exponents
论文作者
论文摘要
令$ f $为有限字段，$μ$为固定添加字符，$ s $是$ | f^\ times | $的整数。对于f $中的任何$ a \，相应的weil和定义为$ w_ {f，s}（a）= \ displaystyle \ sum_ {x \ in F}μ（x^s-ax）$。 Weil频谱将Weil Sum的不同值计算为$ a $通过有限字段中的可逆元素运行。这些总和的价值和Weil频谱的大小特别令人感兴趣，因为它们将编码和信息理论中的问题与数学理论和算术几何形状联系起来。在NIHO指数的设置中，我们检查了Weil和其界限及其频谱。结果，我们在NIHO指数的情况下，对Helleseth（$ 1971美元）的消失猜想（1971美元）给出了新的证据。我们还指出了何时至少包含五个元素，并证明了一定类别的Weil总和。
Let $F$ be a finite field, $μ$ be a fixed additive character and $s$ be an integer coprime to $|F^\times|$. For any $a\in F$, the corresponding Weil sum is defined to be $W_{F,s}(a)=\displaystyle\sum_{x \in F} μ(x^s-ax)$. The Weil spectrum counts distinct values of the Weil sum as $a$ runs through the invertible elements in the finite field. The value of these sums and the size of the Weil spectrum are of particular interest, as they link problems in coding and information theory to other areas of math such as number theory and arithmetic geometry. In the setting of Niho exponents, we examine the Weil sum, its bounds and its spectrum. As a consequence, we give a new proof to the Vanishing Conjecture of Helleseth ($1971$) on the presence of zero in the Weil spectrum in the case of Niho exponents. We also state a conjecture for when the Weil spectrum contains at least five elements and prove it for a certain class of Weil sums.
