A class of quaternary sequences with low correlation

Previous Article
Cyclic orbit codes and stabilizer subfields
AMC Home
This Issue
Next Article
Binary codes from reflexive uniform subset graphs on $3$-sets

May 2015, 9(2): 199-210. doi: 10.3934/amc.2015.9.199

A class of quaternary sequences with low correlation

Nian Li ^1, , Xiaohu Tang ^1, and Tor Helleseth ^2,

1.	Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu, Sichuan 610031, China, China
2.	Department of Informatics, University of Bergen, N-5020 Bergen

Received March 2014 Revised November 2014 Published May 2015

Abstract
Related Papers

A class of quaternary sequences $\mathbb{S}_{\lambda}$ had been proven to be optimal for some special values of $\lambda$. In this note, $\mathbb{S}_{\lambda}$ is investigated for all $\lambda$ by virtue of the $\mathbb{Z}_4$-valued quadratic forms over Galois rings. As a consequence, a new class of quaternary sequences with low correlation is obtained and the correlation distribution is also completely determined. It also turns out that the known optimal quaternary sequences $\mathbb{S}_{\lambda}$ for particular $\lambda$ can be easily obtained from our approach.

Keywords: Galois ring, quaternary sequence, quadratic form, low correlation..

Mathematics Subject Classification: Primary: 94A05, 94A5.

Citation: Nian Li, Xiaohu Tang, Tor Helleseth. A class of quaternary sequences with low correlation. Advances in Mathematics of Communications, 2015, 9 (2) : 199-210. doi: 10.3934/amc.2015.9.199

References:

[1]	S. Boztas, R. Hammons and P. V. Kumar, $4$-phase sequences with near-optimum correlation properties,, IEEE Trans. Inf. Theory, 14 (1992), 1101. doi: 10.1109/18.135649. Google Scholar
[2]	S. Boztas and P. V. Kumar, Binary sequences with Gold-like correlation but larger linear span,, IEEE Trans. Inf. Theory, 40 (1994), 532. doi: 10.1109/18.312181. Google Scholar
[3]	E. H. Brown, Generalizations of the Kervaire invariant,, Annals Math., 95 (1972), 368. doi: 10.2307/1970804. Google Scholar
[4]	P. Fan and M. Darnell, Sequence Design for Communications Applications,, John Wiley, (1996). Google Scholar
[5]	R. Gold, Maximal recursive sequences with $3$-valued recursive crosscorrelation functions,, IEEE Trans. Inf. Theory, 14 (1968), 154. doi: 10.1109/TIT.1968.1054106. Google Scholar
[6]	T. Helleseth and P. V. Kumar, Sequences with low correlation,, in Handbook of Coding Theory* (eds. V.S. Pless and W.C. Huffman)*, (1998). Google Scholar
[7]	W. Jiang, L. Hu, X. Tang and X. Zeng, New optimal quadriphase sequences with larger linear span,, IEEE Trans. Inf. Theory, 55 (2009), 458. doi: 10.1109/TIT.2008.2008122. Google Scholar
[8]	A. Johansen, T. Helleseth and X. Tang, The correlation distribution of quaternary sequences of period $2(2^n-1)$,, IEEE Trans. Inf. Theory, 54 (2008), 3130. doi: 10.1109/TIT.2008.924727. Google Scholar
[9]	T. Kasami, Weight Distribution Formula for Some Class of Cyclic Codes,, Coordinated Sci. Lab., (1966). Google Scholar
[10]	S. H. Kim and J. S. No, New families of binary sequences with low crosscorrelation property,, IEEE Trans. Inf. Theory, 49 (2003), 3059. doi: 10.1109/TIT.2003.818399. Google Scholar
[11]	P. V. Kumar, T. Helleseth, A. R. Calderbank and A. R. Hammons, Large families of quaternary sequences with low correlation,, IEEE Trans. Inf. Theory, 42 (1996), 579. doi: 10.1109/18.485726. Google Scholar
[12]	N. Li, X. Tang, X. Zeng and L. Hu, On the correlation distributions of optimal quaternary sequence family $\mathcal U$ and optimal binary sequence family $\mathcal V$,, IEEE Trans. Inf. Theory, 57 (2011), 3815. doi: 10.1109/TIT.2011.2132670. Google Scholar
[13]	K.-U. Schmidt, $\mathbb Z_4$-valued quadratic forms and quaternary sequence families,, IEEE Trans. Inf. Theory, 55 (2009), 5803. doi: 10.1109/TIT.2009.2032818. Google Scholar
[14]	V. Sidelnikov, On mutual correlation of sequences,, Soviet Math. Dokl., 12 (1971), 197. Google Scholar
[15]	X. Tang and T. Helleseth, Generic construction of quaternary sequences of period $2N$ with low correlation from quaternary sequences of odd period $N$,, IEEE Trans. Inf. Theory, 57 (2011), 2295. doi: 10.1109/TIT.2011.2110290. Google Scholar
[16]	X. Tang, T. Helleseth and P. Fan, A new optimal quaternary sequence family of length $2(2^n-1)$ obtained from the orthogonal transformation of families $\mathcal B$ and $\mathcal C$,, Des. Codes Crypt., 53 (2009), 137. doi: 10.1007/s10623-009-9294-y. Google Scholar
[17]	X. Tang, T. Helleseth, L. Hu and W. Jiang, Two new families of optimal binary sequences obtained from quaternary sequences,, IEEE Trans. Inf. Theory, 55 (2009), 433. doi: 10.1109/TIT.2009.2013023. Google Scholar
[18]	X. Tang and P. Udaya, A note on the optimal quadriphase sequences families,, IEEE Trans. Inf. Theory, 53 (2007), 433. doi: 10.1109/TIT.2006.887502. Google Scholar
[19]	X. Tang, P. Udaya and P. Fan, Quadriphase sequences obtained from binary quadratic form sequences,, in Sequences and Their Applications - SETA 2004, (2004), 243. doi: 10.1007/11423461_17. Google Scholar
[20]	P. Udaya, Polyphase and Frequency Hopping Sequences Obtained from Finite Rings,, Ph.D thesis, (1992). Google Scholar
[21]	P. Udaya and M. U. Siddiqi, Optimal and suboptimal quadriphase sequences derived from maximal length sequences over $\mathbb Z_4$,, Appl. Algebra Eng. Commun. Comput., 9 (1998), 161. doi: 10.1007/s002000050101. Google Scholar
[22]	L. R. Welch, Lower bounds on the maximum crosscorrelation on the signals,, IEEE Trans. Inf. Theory, 20 (1974), 397. doi: 10.1109/TIT.1974.1055219. Google Scholar

show all references

References:

[1]	S. Boztas, R. Hammons and P. V. Kumar, $4$-phase sequences with near-optimum correlation properties,, IEEE Trans. Inf. Theory, 14 (1992), 1101. doi: 10.1109/18.135649. Google Scholar
[2]	S. Boztas and P. V. Kumar, Binary sequences with Gold-like correlation but larger linear span,, IEEE Trans. Inf. Theory, 40 (1994), 532. doi: 10.1109/18.312181. Google Scholar
[3]	E. H. Brown, Generalizations of the Kervaire invariant,, Annals Math., 95 (1972), 368. doi: 10.2307/1970804. Google Scholar
[4]	P. Fan and M. Darnell, Sequence Design for Communications Applications,, John Wiley, (1996). Google Scholar
[5]	R. Gold, Maximal recursive sequences with $3$-valued recursive crosscorrelation functions,, IEEE Trans. Inf. Theory, 14 (1968), 154. doi: 10.1109/TIT.1968.1054106. Google Scholar
[6]	T. Helleseth and P. V. Kumar, Sequences with low correlation,, in Handbook of Coding Theory* (eds. V.S. Pless and W.C. Huffman)*, (1998). Google Scholar
[7]	W. Jiang, L. Hu, X. Tang and X. Zeng, New optimal quadriphase sequences with larger linear span,, IEEE Trans. Inf. Theory, 55 (2009), 458. doi: 10.1109/TIT.2008.2008122. Google Scholar
[8]	A. Johansen, T. Helleseth and X. Tang, The correlation distribution of quaternary sequences of period $2(2^n-1)$,, IEEE Trans. Inf. Theory, 54 (2008), 3130. doi: 10.1109/TIT.2008.924727. Google Scholar
[9]	T. Kasami, Weight Distribution Formula for Some Class of Cyclic Codes,, Coordinated Sci. Lab., (1966). Google Scholar
[10]	S. H. Kim and J. S. No, New families of binary sequences with low crosscorrelation property,, IEEE Trans. Inf. Theory, 49 (2003), 3059. doi: 10.1109/TIT.2003.818399. Google Scholar
[11]	P. V. Kumar, T. Helleseth, A. R. Calderbank and A. R. Hammons, Large families of quaternary sequences with low correlation,, IEEE Trans. Inf. Theory, 42 (1996), 579. doi: 10.1109/18.485726. Google Scholar
[12]	N. Li, X. Tang, X. Zeng and L. Hu, On the correlation distributions of optimal quaternary sequence family $\mathcal U$ and optimal binary sequence family $\mathcal V$,, IEEE Trans. Inf. Theory, 57 (2011), 3815. doi: 10.1109/TIT.2011.2132670. Google Scholar
[13]	K.-U. Schmidt, $\mathbb Z_4$-valued quadratic forms and quaternary sequence families,, IEEE Trans. Inf. Theory, 55 (2009), 5803. doi: 10.1109/TIT.2009.2032818. Google Scholar
[14]	V. Sidelnikov, On mutual correlation of sequences,, Soviet Math. Dokl., 12 (1971), 197. Google Scholar
[15]	X. Tang and T. Helleseth, Generic construction of quaternary sequences of period $2N$ with low correlation from quaternary sequences of odd period $N$,, IEEE Trans. Inf. Theory, 57 (2011), 2295. doi: 10.1109/TIT.2011.2110290. Google Scholar
[16]	X. Tang, T. Helleseth and P. Fan, A new optimal quaternary sequence family of length $2(2^n-1)$ obtained from the orthogonal transformation of families $\mathcal B$ and $\mathcal C$,, Des. Codes Crypt., 53 (2009), 137. doi: 10.1007/s10623-009-9294-y. Google Scholar
[17]	X. Tang, T. Helleseth, L. Hu and W. Jiang, Two new families of optimal binary sequences obtained from quaternary sequences,, IEEE Trans. Inf. Theory, 55 (2009), 433. doi: 10.1109/TIT.2009.2013023. Google Scholar
[18]	X. Tang and P. Udaya, A note on the optimal quadriphase sequences families,, IEEE Trans. Inf. Theory, 53 (2007), 433. doi: 10.1109/TIT.2006.887502. Google Scholar
[19]	X. Tang, P. Udaya and P. Fan, Quadriphase sequences obtained from binary quadratic form sequences,, in Sequences and Their Applications - SETA 2004, (2004), 243. doi: 10.1007/11423461_17. Google Scholar
[20]	P. Udaya, Polyphase and Frequency Hopping Sequences Obtained from Finite Rings,, Ph.D thesis, (1992). Google Scholar
[21]	P. Udaya and M. U. Siddiqi, Optimal and suboptimal quadriphase sequences derived from maximal length sequences over $\mathbb Z_4$,, Appl. Algebra Eng. Commun. Comput., 9 (1998), 161. doi: 10.1007/s002000050101. Google Scholar
[22]	L. R. Welch, Lower bounds on the maximum crosscorrelation on the signals,, IEEE Trans. Inf. Theory, 20 (1974), 397. doi: 10.1109/TIT.1974.1055219. Google Scholar

[1]	Hua Liang, Wenbing Chen, Jinquan Luo, Yuansheng Tang. A new nonbinary sequence family with low correlation and large size. Advances in Mathematics of Communications, 2017, 11 (4) : 671-691. doi: 10.3934/amc.2017049
[2]	Limengnan Zhou, Daiyuan Peng, Hongyu Han, Hongbin Liang, Zheng Ma. Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation. Advances in Mathematics of Communications, 2018, 12 (1) : 67-79. doi: 10.3934/amc.2018004
[3]	Claude Carlet, Juan Carlos Ku-Cauich, Horacio Tapia-Recillas. Bent functions on a Galois ring and systematic authentication codes. Advances in Mathematics of Communications, 2012, 6 (2) : 249-258. doi: 10.3934/amc.2012.6.249
[4]	Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim, Dae-Woon Lim. New construction methods of quaternary periodic complementary sequence sets. Advances in Mathematics of Communications, 2010, 4 (1) : 61-68. doi: 10.3934/amc.2010.4.61
[5]	Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim. New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set. Advances in Mathematics of Communications, 2009, 3 (2) : 115-124. doi: 10.3934/amc.2009.3.115
[6]	Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475
[7]	Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55
[8]	Fanxin Zeng, Xiaoping Zeng, Zhenyu Zhang, Guixin Xuan. Quaternary periodic complementary/Z-complementary sequence sets based on interleaving technique and Gray mapping. Advances in Mathematics of Communications, 2012, 6 (2) : 237-247. doi: 10.3934/amc.2012.6.237
[9]	Jacinto Marabel Romo. A closed-form solution for outperformance options with stochastic correlation and stochastic volatility. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1185-1209. doi: 10.3934/jimo.2015.11.1185
[10]	Wenbing Chen, Jinquan Luo, Yuansheng Tang, Quanquan Liu. Some new results on cross correlation of $p$-ary $m$-sequence and its decimated sequence. Advances in Mathematics of Communications, 2015, 9 (3) : 375-390. doi: 10.3934/amc.2015.9.375
[11]	Zhenyu Zhang, Lijia Ge, Fanxin Zeng, Guixin Xuan. Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties. Advances in Mathematics of Communications, 2015, 9 (1) : 9-21. doi: 10.3934/amc.2015.9.9
[12]	Xiaohui Liu, Jinhua Wang, Dianhua Wu. Two new classes of binary sequence pairs with three-level cross-correlation. Advances in Mathematics of Communications, 2015, 9 (1) : 117-128. doi: 10.3934/amc.2015.9.117
[13]	Yuhua Sun, Zilong Wang, Hui Li, Tongjiang Yan. The cross-correlation distribution of a $p$-ary $m$-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$. Advances in Mathematics of Communications, 2013, 7 (4) : 409-424. doi: 10.3934/amc.2013.7.409
[14]	Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001
[15]	Hua Liang, Jinquan Luo, Yuansheng Tang. On cross-correlation of a binary $m$-sequence of period $2^{2k}-1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$. Advances in Mathematics of Communications, 2017, 11 (4) : 693-703. doi: 10.3934/amc.2017050
[16]	Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563
[17]	Jing Zhou, Zhibin Deng. A low-dimensional SDP relaxation based spatial branch and bound method for nonconvex quadratic programs. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2019044
[18]	Lassi Roininen, Markku S. Lehtinen, Sari Lasanen, Mikko Orispää, Markku Markkanen. Correlation priors. Inverse Problems & Imaging, 2011, 5 (1) : 167-184. doi: 10.3934/ipi.2011.5.167
[19]	Stefanella Boatto. Curvature perturbations and stability of a ring of vortices. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 349-375. doi: 10.3934/dcdsb.2008.10.349
[20]	Ferenc Szöllősi. On quaternary complex Hadamard matrices of small orders. Advances in Mathematics of Communications, 2011, 5 (2) : 309-315. doi: 10.3934/amc.2011.5.309

2018 Impact Factor: 0.879

American Institute of Mathematical Sciences