American Institute of Mathematical Sciences

Advanced
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

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), 1101-1113. 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), 532-537. doi: 10.1109/18.312181.  Google Scholar

[3]

E. H. Brown, Generalizations of the Kervaire invariant, Annals Math., 95 (1972), 368-383. 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), 154-156. 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), Elsevier, Amsterdman, 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), 458-470. 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), 3130-3139. doi: 10.1109/TIT.2008.924727.  Google Scholar

[9]

T. Kasami, Weight Distribution Formula for Some Class of Cyclic Codes, Coordinated Sci. Lab., Univ. Illinois Urbana-Champaign, Tech. Rep. R-285, 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), 3059-3065. 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), 579-592. 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), 3815-3824. 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), 5803-5810. doi: 10.1109/TIT.2009.2032818.  Google Scholar

[14]

V. Sidelnikov, On mutual correlation of sequences, Soviet Math. Dokl., 12 (1971), 197-201.  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), 2295-2300. 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), 137-148. 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), 433-436. 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), 433-436. 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, 2005, 243-254. doi: 10.1007/11423461_17.  Google Scholar

[20]

P. Udaya, Polyphase and Frequency Hopping Sequences Obtained from Finite Rings, Ph.D thesis, Dept. Elec. Eng., Indian Inst. Technol., Kanpur, 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), 161-191. 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), 397-399. 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), 1101-1113. 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), 532-537. doi: 10.1109/18.312181.  Google Scholar

[3]

E. H. Brown, Generalizations of the Kervaire invariant, Annals Math., 95 (1972), 368-383. 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), 154-156. 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), Elsevier, Amsterdman, 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), 458-470. 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), 3130-3139. doi: 10.1109/TIT.2008.924727.  Google Scholar

[9]

T. Kasami, Weight Distribution Formula for Some Class of Cyclic Codes, Coordinated Sci. Lab., Univ. Illinois Urbana-Champaign, Tech. Rep. R-285, 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), 3059-3065. 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), 579-592. 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), 3815-3824. 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), 5803-5810. doi: 10.1109/TIT.2009.2032818.  Google Scholar

[14]

V. Sidelnikov, On mutual correlation of sequences, Soviet Math. Dokl., 12 (1971), 197-201.  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), 2295-2300. 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), 137-148. 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), 433-436. 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), 433-436. 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, 2005, 243-254. doi: 10.1007/11423461_17.  Google Scholar

[20]

P. Udaya, Polyphase and Frequency Hopping Sequences Obtained from Finite Rings, Ph.D thesis, Dept. Elec. Eng., Indian Inst. Technol., Kanpur, 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), 161-191. 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), 397-399. 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]

Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, 2020, 14 (2) : 319-332. doi: 10.3934/amc.2020023
[5]

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
[6]

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
[7]

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
[8]

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
[9]

Ferruh Özbudak, Eda Tekin. Correlation distribution of a sequence family generalizing some sequences of Trachtenberg. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020087
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

Huaning Liu, Xi Liu. On the correlation measures of orders $ 3 $ and $ 4 $ of binary sequence of period $ p^2 $ derived from Fermat quotients. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021008
[16]

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
[17]

Hongyu Han, Sheng Zhang. New classes of strictly optimal low hit zone frequency hopping sequence sets. Advances in Mathematics of Communications, 2020, 14 (4) : 579-589. doi: 10.3934/amc.2020031
[18]

Wenjuan Yin, Can Xiang, Fang-Wei Fu. Two constructions of low-hit-zone frequency-hopping sequence sets. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020110
[19]

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
[20]

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

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (152)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences