    doi: 10.3934/amc.2020112

## On the equivalence of several classes of quaternary sequences with optimal autocorrelation and length $2p$

 1 Fujian Provincial Key Laboratory of Network Security and Cryptology, College of Mathematics and Informatics, Fujian Normal University, Fuzhou, Fujian 350117, China 2 School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China

* Corresponding author: Pinhui Ke

Received  April 2020 Revised  July 2020 Published  September 2020

Fund Project: The authors are supported by National Natural Science Foundation of China (No. 61772292, 61772476, 61771016), Natural Science Foundation of Fujian Province (No. 2019J01273) and Fujian Normal University Innovative Research Team (IRTL1207)

Quaternary sequences with optimal autocorrelation property are preferred in applications. Cyclotomic classes of order 4 are widely used in the constructions of quaternary sequences due to the convenience of defining a quaternary sequence with the cyclotomic classes of order 4 as its support set. Recently, several classes of optimal quaternary sequences of period $2p$, which are all closely related to the cyclotomic classes of order 4 with respect to $\mathbb{Z}_p$ were introduced in the literature. However, less attention has been paid to the equivalence between these known results. In this paper, we introduce the unified form of this kind of quaternary sequences to classify these known results and then conclude the unified forms of these optimal quaternary sequences. By doing this, we disclose the relationship between the optimal quaternary sequences derived from different methods in the literature on one hand. And on the other hand, when the new obtained optimal quaternary sequence period is $2p$ and the cyclotomic classes of order 4 are involved, the methods and the results given in this paper can be used to identify if the sequence is new or not.

Citation: Pinhui Ke, Panpan Qiao, Yang Yang. On the equivalence of several classes of quaternary sequences with optimal autocorrelation and length $2p$. Advances in Mathematics of Communications, doi: 10.3934/amc.2020112
##### References:
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##### References:
  Y. Cai and C. Ding, Binary sequences with optimal autocorrelation, Theoret. Comput. Sci., 410 (2009), 2316-2322.  doi: 10.1016/j.tcs.2009.02.021.  Google Scholar  J. Chung, Y. K. Han and K. Yang, New quaternary sequences with even period and three-valued autocorrelation, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., E93-A(1) (2010), 309-315.  doi: 10.1587/transfun.E93.A.309. Google Scholar  C. Ding, T. Helleseth and H. M. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inf. Theory, 47 (2001), 428-433.  doi: 10.1109/18.904555.  Google Scholar  V. Edemskiy and A. Ivanov, Autocorrelation and linear complexity of quaternary sequences of period $2p$ based on cyclotomic classes of order four, 2013 IEEE International Symposium on Information Theory, (2013), 3120–3124. doi: 10.1109/ISIT.2013.6620800. Google Scholar  G. Golomb and G. Gong, Signal Designs with Good Correlations: For Wireless Communications, Cryptography and Radar Applications, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511546907.  Google Scholar  D. H. Green and P. R. Green, Polyphase-related prime sequences, IEEE Proc. Comput. Digit. Tech., 148 (2001), 53-62.  doi: 10.1049/ip-cdt:20010209. Google Scholar  J. W. Jang, Y. S. Kim, S. H. Kim and J. S. No, New quaternary sequences with ideal autocorrelation constructed from binary sequences with ideal autocorrelation, Proc. ISIT 2009, Seoul, Korea, (2009), 278–281. doi: 10.1109/ISIT.2009.5205807. Google Scholar  Y. S. Kim, J. W. Jang, S. H. Kim and J. S. No, New construction of quaternary sequences with ideal autocorrelation from Legendre sequences, Proc. ISIT 2009, Seoul, Korea, (2009), 282–285. doi: 10.1109/ISIT.2009.5205767. Google Scholar  Y. S. Kim, J. W. Jang, S. H. Kim and J. S. No, New quaternary sequences with optimal autocorrelation, Proc. ISIT 2009, Seoul, Korea, (2009), 286–289. Google Scholar  N. Li, X. H. Tang and T. Helleseth, New $M$-ary sequences with low autocorrelation from interleaved technique, Des. Codes Cryptogr., 73 (2014), 237-249.  doi: 10.1007/s10623-013-9821-8.  Google Scholar  L. F. Luo and W. P. Ma, Balanced quaternary sequences of even period with optimal autocorrelation, IET Commun., 13 (2019), 1808-1812.  doi: 10.1049/iet-com.2018.5192. Google Scholar  J. Michel and Q. Wang, Some new balanced and almost balanced quaternary sequences with low autocorrelation, Cryptogr. Commun., 11 (2019), 191-206.  doi: 10.1007/s12095-018-0281-x.  Google Scholar  H. Schotten, Optimum complementary sets and quadriphase sequences derived from q-ary m-sequences, Proc. IEEE Int. Symp. Inf. Theory, Ulm, Germany, (1997), 485. doi: 10.1109/ISIT.1997.613422. Google Scholar  X. Shen, Y. Jia, J. Wang and L. Zhang, New families of balanced quaternary sequences of even period with three-level optimal autocorrelation, IEEE Commun. Lett., 21 (2017), 2146-2149.  doi: 10.1109/LCOMM.2017.2661750. Google Scholar  V. M. Sidelnikov, Some $k$-vauled pseudo-random sequences and nearly equidistant codes, Probl. Inf. Trans., 5 (1969), 12-16. Google Scholar  T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics. Markham Publishing Co., Chicago, 1967. Google Scholar  W. Su, et al., New quaternary sequences of even length with optimal autocorrelation, Sci. China (Inf. Sci.), 61 (2018), 022308-1-022308-131. doi: 10.1007/s11432-016-9087-2.  Google Scholar  X. H. Tang and J. Lindner, Almost quadriphase sequence with ideal autocorrelation property, IEEE Signal Proc. Lett., 16 (2009), 38-40.   Google Scholar  X. H. Tang and C. Ding, New classes of balanced quaternary and almost balanced binary sequences with optimal autocorrelation value, IEEE Trans. Inf. Theory, 56 (2010), 6398-6405.  doi: 10.1109/TIT.2010.2081170.  Google Scholar  Z. Yang and P. H. Ke, Quaternary sequences with odd period and low autocorrelation, Elec. Lett., 46 (2010), 1068-1069.  doi: 10.1049/el.2010.1685. Google Scholar  Z. Yang and P. H. Ke, Construction of quaternary sequences of length pq with low autocorrelation, Cryptogr. Commun., 3 (2011), 55-64.  doi: 10.1007/s12095-010-0034-y.  Google Scholar
Known Optimal Quaternary Sequences of Length $2p$
 Sequence Construction (Sketch) Constrains Chung et al.   $\mathbf{s}=\phi^{-1}(\mathbf{a}, L^{p}(\mathbf{b}))$ $\mathbf{b}=\mathbf{a}$ or $\mathbf{a}+1$ $p\equiv 5\pmod 8$ $\mathbf{a}$ is DHM sequence of length $p$ Su et al.  $\mathbf{s}=\phi^{-1}(\mathbf{c}, \mathbf{d}))$ $\mathbf{c}=I(\mathbf{a}_{0}, e(0)+L^{\lambda}(\mathbf{a}_{1}))$ $\mathbf{d}=I(e(1)+\mathbf{a}_{2}, e(2)+L^{\lambda}(\mathbf{a}_{3}))$ $p\equiv 1\pmod 4$, $\lambda=\frac{p+1}{2}$ $(e(0), e(1), e(2))\in \mathbb{Z}_2^3$ $\mathbf{a}_i\in\{\mathbf{s_1}, \mathbf{s_2}, \cdots, \mathbf{s_6}\}, 0\le i\le 3$ Shen et al.  $\mbox{Supp}_{\mathbf{s}}(0)=\psi(\{0\}\times D_{i_0}\cup \{1\}\times D_{j_0})\cup \{0\}$ $\mbox{Supp}_{\mathbf{s}}(2)=\psi(\{0\}\times D_{i_2}\cup \{1\}\times D_{j_2})\cup \{p\}$ $\mbox{Supp}_{\mathbf{s}}(t)=\psi(\{0\}\times D_{i_t}\cup \{1\}\times D_{j_t}), \ \mbox{for} \ t=1, 3$ $p\equiv 1\pmod 4$ $i_l\neq i_m$ and $j_l\neq j_m$, if $l\neq m$ Luo et al.  $\mathbf{s}=I(\mathbf{u}_c, L^{\lambda}(\mathbf{v}_c)+2)$ $u_c(0)$ and $u_c(i)=j, \mbox{if}\ i\in D_{(m+j)\pmod 4}$ $v_c(0)=c$ and $v_c(i)=j, \mbox{if}\ i\in D_{(n+3j)\pmod 4}$ $p\equiv 1\pmod 4$, $\lambda=\frac{p+1}{2}$ $m, n, c\in \mathbb{Z}_4$ $f$ is odd and $m-n\equiv 2c\pmod 4$ or $f$ is even and $m-n\equiv 2c+2\pmod 4$
 Sequence Construction (Sketch) Constrains Chung et al.   $\mathbf{s}=\phi^{-1}(\mathbf{a}, L^{p}(\mathbf{b}))$ $\mathbf{b}=\mathbf{a}$ or $\mathbf{a}+1$ $p\equiv 5\pmod 8$ $\mathbf{a}$ is DHM sequence of length $p$ Su et al.  $\mathbf{s}=\phi^{-1}(\mathbf{c}, \mathbf{d}))$ $\mathbf{c}=I(\mathbf{a}_{0}, e(0)+L^{\lambda}(\mathbf{a}_{1}))$ $\mathbf{d}=I(e(1)+\mathbf{a}_{2}, e(2)+L^{\lambda}(\mathbf{a}_{3}))$ $p\equiv 1\pmod 4$, $\lambda=\frac{p+1}{2}$ $(e(0), e(1), e(2))\in \mathbb{Z}_2^3$ $\mathbf{a}_i\in\{\mathbf{s_1}, \mathbf{s_2}, \cdots, \mathbf{s_6}\}, 0\le i\le 3$ Shen et al.  $\mbox{Supp}_{\mathbf{s}}(0)=\psi(\{0\}\times D_{i_0}\cup \{1\}\times D_{j_0})\cup \{0\}$ $\mbox{Supp}_{\mathbf{s}}(2)=\psi(\{0\}\times D_{i_2}\cup \{1\}\times D_{j_2})\cup \{p\}$ $\mbox{Supp}_{\mathbf{s}}(t)=\psi(\{0\}\times D_{i_t}\cup \{1\}\times D_{j_t}), \ \mbox{for} \ t=1, 3$ $p\equiv 1\pmod 4$ $i_l\neq i_m$ and $j_l\neq j_m$, if $l\neq m$ Luo et al.  $\mathbf{s}=I(\mathbf{u}_c, L^{\lambda}(\mathbf{v}_c)+2)$ $u_c(0)$ and $u_c(i)=j, \mbox{if}\ i\in D_{(m+j)\pmod 4}$ $v_c(0)=c$ and $v_c(i)=j, \mbox{if}\ i\in D_{(n+3j)\pmod 4}$ $p\equiv 1\pmod 4$, $\lambda=\frac{p+1}{2}$ $m, n, c\in \mathbb{Z}_4$ $f$ is odd and $m-n\equiv 2c\pmod 4$ or $f$ is even and $m-n\equiv 2c+2\pmod 4$
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