特征线性组合的低相关序列
Low Correlation Sequences from Linear Combinations of Characters
摘要：
使用有限域的乘法字符的线性组合形成的二进制序列对展示，相比随机序列对，同时实现显着较低的均方自相关值（对中的每个序列）和显着较低的均方互相关值。如果我们类似于自相关的通常优值因子定义互相关优值因子，并且如果我们将劣值因子定义为优值因子的倒数，则已知随机选择的二进制序列对具有平均互相关劣值因子$1$。我们的构造提供了具有显著小于$1$的互相关缺陷因子的序列对，并且同时，也可以使各个序列的自相关缺陷因子显著小于$1$（这也表明优于平均性能）。这里研究的序列对提供了自相关和互相关性能的组合，这是使用由单个字符形成的序列无法实现的，例如最大线性递归序列（m序列）和勒让德序列。在这项研究中，精确的渐近公式证明的自相关和互相关的价值因子的序列对形成的乘法字符的线性组合。数据表明，渐近行为是密切接近的序列长度适中。
英文摘要：
Pairs of binary sequences formed using linear combinations of multiplicative characters of finite fields are exhibited that, when compared to random sequence pairs, simultaneously achieve significantly lower mean square autocorrelation values (for each sequence in the pair) and significantly lower mean square crosscorrelation values. If we define crosscorrelation merit factor analogously to the usual merit factor for autocorrelation, and if we define demerit factor as the reciprocal of merit factor, then randomly selected binary sequence pairs are known to have an average crosscorrelation demerit factor of $1$. Our constructions provide sequence pairs with crosscorrelation demerit factor significantly less than $1$, and at the same time, the autocorrelation demerit factors of the individual sequences can also be made significantly less than $1$ (which also indicates better than average performance). The sequence pairs studied here provide combinations of autocorrelation and crosscorrelation performance that are not achievable using sequences formed from single characters, such as maximal linear recursive sequences (m-sequences) and Legendre sequences. In this study, exact asymptotic formulae are proved for the autocorrelation and crosscorrelation merit factors of sequence pairs formed using linear combinations of multiplicative characters. Data is presented that shows that the asymptotic behavior is closely approximated by sequences of modest length.
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