论文标题
二进制序列来自连续二次残基的差异
Binary sequences derived from differences of consecutive quadratic residues
论文作者
论文摘要
对于prime $ p \ ge 5 $ let $ q_0，q_1，\ ldots，q _ {（p-3）/2} $是二次残基Modulo $ p $，以增加顺序。我们研究两个$（p-3）/2 $ - 周期性二进制序列$（d_n）$和$（t_n）$由$ d_n = q_n = q_n+q_n+q_ {n+1} \ bmod 2 $ and $ t_n = 1 $，如果$ q_ {n+1} = q_n+1} = q_n+1 $和$ t_n = 0 $否则，否则$ n = 0,1，\ ldots，（p-5）/2 $。对于这两个序列，我们都会发现一些足够的条件来达到最大线性复杂性$（P-3）/2 $。研究$（d_n）$的线性复杂性是由Caragiu等人的启发式动机的。但是，$（d_n）$无法平衡，我们表明$（d_n）$的周期约为$ 1/3 $零，如果$ p $足够大，则包含$ 2/3 $。相反，$（t_n）$本质上是平衡的，而且所有更长的长度$ s $的模式在矢量序列$（t_n，t_n，t_n，t_ {n+1}，\ ldots，t_ {n+s-1}）中，t_ {n+s-1} $，$ n = 0,1，\ n = 0,1，\ ldots，\ ldots，（p-5 $ p $ p $ s $ s $ s $）
For a prime $p\ge 5$ let $q_0,q_1,\ldots,q_{(p-3)/2}$ be the quadratic residues modulo $p$ in increasing order. We study two $(p-3)/2$-periodic binary sequences $(d_n)$ and $(t_n)$ defined by $d_n=q_n+q_{n+1}\bmod 2$ and $t_n=1$ if $q_{n+1}=q_n+1$ and $t_n=0$ otherwise, $n=0,1,\ldots,(p-5)/2$. For both sequences we find some sufficient conditions for attaining the maximal linear complexity $(p-3)/2$. Studying the linear complexity of $(d_n)$ was motivated by heuristics of Caragiu et al. However, $(d_n)$ is not balanced and we show that a period of $(d_n)$ contains about $1/3$ zeros and $2/3$ ones if $p$ is sufficiently large. In contrast, $(t_n)$ is not only essentially balanced but also all longer patterns of length $s$ appear essentially equally often in the vector sequence $(t_n,t_{n+1},\ldots,t_{n+s-1})$, $n=0,1,\ldots,(p-5)/2$, for any fixed $s$ and sufficiently large $p$.
