Four-valued cross-correlation found between sequences with different periods
A class of Niho type four-valued cross-correlation functions between sequences of different periods
Information Theory
Summary
Understanding how certain repeating sequences interact is important for communication and data processing. This paper shows that when you look at a specific kind of binary sequence and a related shorter sequence created from it, their interaction has exactly four possible values. The authors provide a full description of these values and how often each occurs. Their result also includes some previously known cases as special examples.
What this means in practice
- •For telecommunication engineers: Design communication systems using sequences with known cross-correlation to improve signal separation and reduce interference.
- •For cryptography developers: Use sequences with controlled cross-correlation properties to enhance the security of cryptographic keys and random number generation.
Authors
Yao Yao, Yufei Wang, Yongbo Xia, Nian Li, Xiangyong Zeng
Abstract
Determining the cross-correlation distributions between $m$-sequences and their decimated sequences is an important problem in sequence design. For two positive integers $n$ and $k$ with $n=2k$, this paper investigates the cross-correlation function between a binary $m$-sequence $\{u(t)\}$ of period $2^{n}-1$ and its $2^{\gcd(s,k)}+1$ short-period decimated sequences $\{u(dt+l)\}$, where the decimation $d=\frac{2^s}{2^s+1}(2^k-1)+1$ satisfies $1<s<n$ and $v_2(s)>v_2(k)$. The cross-correlation function is proved to be four-valued and its value distribution is completely determined. Several previously known binary cross-correlation functions are included in our result as special cases.