论文标题
关于Costas排列的互相关的注释
A Note on the Cross-Correlation of Costas Permutations
论文作者
论文摘要
我们以Drakakis等人的工作为基础。 （2011年）关于韦尔奇和戈洛姆·科斯塔斯置换家族的最大互相关。特别是，我们解决了他们的一些猜想。更确切地说，我们证明了两个结果。 首先，对于$φ（p-1）$不同的welch costas $ \ {1，\ ldots，p-1 \} $是$（p-1）/t $的最大互相相关，$ \ {1，\ ldots，p-1 \} $ 1+p^{1/2} $否则。这里$φ$表示Euler的基本功能，如果$（P-1）/2 $也是PREME，则Prime $ P $是安全的。 其次，对于主要功率，$ q \ ge 4 $最大的golomb costas costas排列$ \ \ {1，\ ldots，q-2 \} $是$（q-1）/t-1 $如果$（q-1）/t-1 $是$（q-1）/2 $ q $ q $ q $ q是$ q $ q- 1 $ Q-1 $不是Prime，最多$ 1+Q^{1/2} $否则。请注意，我们认为一个比Drakakis等人更小的家庭。我们的家庭尺寸为$φ（Q-1）$，而$φ（Q-1）^2 $不同的Golomb Costas排列。 Drakakis等人表中给出的较大家族的最大互相关。对于$ Q $，比我们的界限（对于较小的家庭而言）大。
We build on the work of Drakakis et al. (2011) on the maximal cross-correlation of the families of Welch and Golomb Costas permutations. In particular, we settle some of their conjectures. More precisely, we prove two results. First, for a prime $p\ge 5$, the maximal cross-correlation of the family of the $φ(p-1)$ different Welch Costas permutations of $\{1,\ldots,p-1\}$ is $(p-1)/t$, where $t$ is the smallest prime divisor of $(p-1)/2$ if $p$ is not a safe prime and at most $1+p^{1/2}$ otherwise. Here $φ$ denotes Euler's totient function and a prime $p$ is a safe prime if $(p-1)/2$ is also prime. Second, for a prime power $q\ge 4$ the maximal cross-correlation of a subfamily of Golomb Costas permutations of $\{1,\ldots,q-2\}$ is $(q-1)/t-1$ if $t$ is the smallest prime divisor of $(q-1)/2$ if $q$ is odd and of $q-1$ if $q$ is even provided that $(q-1)/2$ and $q-1$ are not prime, and at most $1+q^{1/2}$ otherwise. Note that we consider a smaller family than Drakakis et al. Our family is of size $φ(q-1)$ whereas there are $φ(q-1)^2$ different Golomb Costas permutations. The maximal cross-correlation of the larger family given in the tables of Drakakis et al. is larger than our bound (for the smaller family) for some $q$.
