论文标题
避免直角和某些锤击距离
Avoiding right angles and certain Hamming distances
论文作者
论文摘要
在本文中，我们表明，$ \ mathbb {f} _q^n $的最大可能大小避免了正确的角度，即不同的向量$ x，y，z $，因此$ x-z $和$ y-z $是彼此依赖的$ o（n^{q-2}）。由于naslund \ cite {naslund}，这会改善以前最著名的界限，并反驳了ge和shangguan \ cite {ge}的猜想。还提出了$ n^{q/3} $的下限。 还表明，$ \ mathbb {f} _q^n $的子集避免了具有所有正确角度的三角形的子集，最多可以具有$ o（n^{2q-2}）$。此外，对于子集$ a \ subseteq \ mathbb {f} _q^n $的最大可能大小的最大可能大小的界限给出了$ x-y $对于任何独特的$ x，y \ in $ in $。确切的答案是针对$ q = 3 $和$ n \ equiv 2 \ pmod {3} $确定的。 我们的方法还可以用来绑定二进制代码的最大可能大小，在该代码中，没有两个代码字可以由固定的Prime $ Q $除外。我们的下限和上限渐近地紧密，并且在无限的许多情况下都很尖锐。
In this paper we show that the largest possible size of a subset of $\mathbb{F}_q^n$ avoiding right angles, that is, distinct vectors $x,y,z$ such that $x-z$ and $y-z$ are perpendicular to each other is at most $O(n^{q-2})$. This improves on the previously best known bound due to Naslund \cite{Naslund} and refutes a conjecture of Ge and Shangguan \cite{Ge}. A lower bound of $n^{q/3}$ is also presented. It is also shown that a subset of $\mathbb{F}_q^n$ avoiding triangles with all right angles can have size at most $O(n^{2q-2})$. Furthermore, asymptotically tight bounds are given for the largest possible size of a subset $A\subseteq \mathbb{F}_q^n$ for which $x-y$ is not self-orthogonal for any distinct $x,y\in A$. The exact answer is determined for $q=3$ and $n\equiv 2\pmod {3}$. Our methods can also be used to bound the maximum possible size of a binary code where no two codewords have Hamming distance divisible by a fixed prime $q$. Our lower- and upper bounds are asymptotically tight and both are sharp in infinitely many cases.
