SCIENCE CHINA Information Sciences, Volume 61 , Issue 2 : 022308(2018) https://doi.org/10.1007/s11432-016-9087-2

## New quaternary sequences of even length with optimal auto-correlation

Wei SU 1,4, Yang YANG 2,4,*,
• AcceptedMar 21, 2017
• PublishedOct 11, 2017
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### Abstract

Sequences with low auto-correlation property have been applied in code-division multiple access communication systems, radar and cryptography. Using the inverse Gray mapping, a quaternary sequence of even length $N$ can be obtained from two binary sequences of the same length, which are called component sequences. In this paper, using interleaving method, we present several classes of component sequences from twin-prime sequences pairs or GMW sequences pairs given by Tang and Ding in 2010; or two, three or four binary sequences defined by cyclotomic classes of order $4$. Hence we can obtain new classes of quaternary sequences, which are different from known ones, since known component sequences are constructed from a pair of binary sequences with optimal auto-correlation or Sidelnikov sequences.

### Acknowledgment

The work of Wei SU was supported by National Science Foundation of China (Grant No. 61402377), and in part supported by Open Research Subject of Key Laboratory (Research Base) of Digital Space Security (Grant No. szjj2014-075), and Science and Technology on Communication Security Laboratory (Grant No. 9140C110302150C11004). The work of Yang YANG was supported by National Science Foundation of China (Grants Nos. 61401376, 11571285), and Application Fundamental Research Plan Project of Sichuan Province (Grant No. 2016JY0160). The work of Zhengchun ZHOU and Xiaohu TANG was supported by National Science Foundation of China (Grants Nos. 61672028, 61325005).

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• Table A1   $f$ odd
 $(i,j)$ $0$ $1$ $2$ $3$ $0$ $A$ $B$ $C$ $D$ $1$ $E$ $E$ $D$ $B$ $2$ $A$ $E$ $A$ $E$ $3$ $E$ $D$ $B$ $E$
• Table A2   $f$ even
 $(i,j)$ $0$ $1$ $2$ $3$ $0$ $A$ $B$ $C$ $D$ $1$ $B$ $D$ $E$ $E$ $2$ $C$ $E$ $C$ $E$ $3$ $D$ $E$ $E$ $B$
• Table A3   The auto- and cross-correlation of six binary sequences of period $n=4f+1$, $f$ odd
 $\tau$ $\{0\}$ $D_0$ $D_1$ $D_2$ $D_3$ $R_{s_1}(\tau)$ $n$ $-2y-1$ $2y-1$ $-2y-1$ $2y-1$ $R_{s_2}(\tau)$ $n$ $-3$ $1$ $-3$ $1$ $R_{s_3}(\tau)$ $n$ $2y-1$ $-2y-1$ $2y-1$ $-2y-1$ $R_{s_4}(\tau)$ $n$ $2y-1$ $-2y-1$ $2y-1$ $-2y-1$ $R_{s_5}(\tau)$ $n$ $1$ $-3$ $1$ $-3$ $R_{s_6}(\tau)$ $n$ $-2y-1$ $2y-1$ $-2y-1$ $2y-1$ $R_{s_1,s_2}(\tau)$ $1$ $-x+2y$ $x+2y+2$ $x-2y-2$ $-x-2y$ $R_{s_1,s_3}(\tau)$ $1$ $x$ $-x+2$ $x$ $-x-2$ $R_{s_1,s_4}(\tau)$ $1$ $-x+2$ $x$ $-x-2$ $x$ $R_{s_1,s_5}(\tau)$ $1$ $x-2y+2$ $-x-2y$ $-x+2y$ $x+2y-2$ $R_{s_1,s_6}(\tau)$ $2-n$ $2y+3$ $3-2y$ $2y-1$ $-1-2y$ $R_{s_2,s_3}(\tau)$ $1$ $x+2y-2$ $x-2y+2$ $-x-2y$ $-x+2y$ $R_{s_2,s_4}(\tau)$ $1$ $-x-2y$ $-x+2y$ $x+2y-2$ $x-2y+2$ $R_{s_2,s_5}(\tau)$ $2-n$ $1$ $1$ $1$ $1$ $R_{s_2,s_6}(\tau)$ $1$ $2y-x$ $x+2y+2$ $x-2y-2$ $-x-2y$ $R_{s_3,s_4}(\tau)$ $2-n$ $3-2y$ $2y-1$ $-1-2y$ $3+2y$ $R_{s_3,s_5}(\tau)$ $1$ $x+2y+2$ $x-2y-2$ $-x-2y$ $-x+2y$ $R_{s_3,s_6}(\tau)$ $1$ $-x+2$ $x$ $-x-2$ $x$ $R_{s_4,s_5}(\tau)$ $1$ $-x-2y$ $-x+2y$ $x+2y+2$ $x-2y-2$ $R_{s_4,s_6}(\tau)$ $1$ $x$ $-x+2$ $x$ $-x-2$ $R_{s_5,s_6}(\tau)$ $1$ $x-2y+2$ $-x-2y$ $-x+2y$ $x+2y-2$
• Table A4   The auto- and cross-correlation of six binary sequences of period $n=4f+1$, $f$ even
 $\tau$ $\{0\}$ $D_0$ $D_1$ $D_2$ $D_3$ $R_{s_1}(\tau)$ $n$ $2y-3$ $-3-2y$ $1+2y$ $1-2y$ $R_{s_2}(\tau)$ $n$ $-3$ $1$ $-3$ $1$ $R_{s_3}(\tau)$ $n$ $-2y-3$ $2y+1$ $-2y+1$ $2y-3$ $R_{s_4}(\tau)$ $n$ $-2y+1$ $2y-3$ $-2y-3$ $2y+1$ $R_{s_5}(\tau)$ $n$ $1$ $-3$ $1$ $-3$ $R_{s_6}(\tau)$ $n$ $2y+1$ $-2y+1$ $2y-3$ $-2y-3$ $R_{s_1,s_2}(\tau)$ $1$ $-x+2y-2$ $x+2y$ $x-2y$ $-x-2y+2$ $R_{s_1,s_3}(\tau)$ $1$ $-x-2$ $x$ $-x+2$ $x$ $R_{s_1,s_4}(\tau)$ $1$ $x$ $-x-2$ $x$ $-x+2$ $R_{s_1,s_5}(\tau)$ $1$ $x-2y$ $-x-2y-2$ $-x+2y+2$ $x+2y$ $R_{s_1,s_6}(\tau)$ $2-n$ $1-2y$ $1+2y$ $1-2y$ $1+2y$ $R_{s_2,s_3}(\tau)$ $1$ $-x-2y-2$ $-x+2y+2$ $x+2y$ $x-2y$ $R_{s_2,s_4}(\tau)$ $1$ $x+2y$ $x-2y$ $-x-2y-2$ $-x+2y+2$ $R_{s_2,s_5}(\tau)$ $2-n$ $1$ $1$ $1$ $1$ $R_{s_2,s_6}(\tau)$ $1$ $x-2y$ $-x-2y+2$ $-x+2y-2$ $x+2y$ $R_{s_3,s_4}(\tau)$ $2-n$ $1+2y$ $1-2y$ $1+2y$ $1-2y$ $R_{s_3,s_5}(\tau)$ $1$ $x+2y$ $x-2y$ $-x-2y+2$ $-x+2y-2$ $R_{s_3,s_6}(\tau)$ $1$ $x$ $-x+2$ $x$ $-x-2$ $R_{s_4,s_5}(\tau)$ $1$ $-x-2y+2$ $-x+2y-2$ $x+2y$ $x-2y$ $R_{s_4,s_6}(\tau)$ $1$ $-x+2$ $x$ $-x-2$ $x$ $R_{s_5,s_6}(\tau)$ $1$ $-x+2y+2$ $x+2y$ $x-2y$ $-x-2y-2~$

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